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Journal of Sound and Vibration

Journal of Sound and Vibration 331 (2012) 1127–1142

0022-46

doi:10.1

n Corr

E-m

palacios

journal homepage: www.elsevier.com/locate/jsvi

Local bifurcations of synchronization in self-excited and forcedunidirectionally coupled micromechanical resonators

Suketu Naik a,n, Takashi Hikihara a, Huy Vu b, Antonio Palacios b, Visarath In c,Patrick Longhini c

a Department of Electrical Engineering, Kyoto University, Katsura, Nishikyo, Kyoto 615-8510, Japanb Nonlinear Dynamical Systems Group, Department of Mathematics, San Diego State University, San Diego, CA 92182, USAc Space and Naval Warfare Systems Center Pacific, Code 71730, 53560 Hull Street, San Diego, CA 92152-5001, USA

a r t i c l e i n f o

Article history:

Received 20 February 2011

Received in revised form

8 September 2011

Accepted 12 October 2011

Handling Editor: L.N. Virginforce. Phenomena such as quasiperiodic oscillations and synchronization are shown by

Available online 5 November 2011

0X/$ - see front matter & 2011 Elsevier Ltd.

016/j.jsv.2011.10.024

esponding author.

ail addresses: suketunaik@gmail.com (S. Naik

@euler.sdsu.edu (A. Palacios), visarath@spaw

a b s t r a c t

A unidirectionally coupled system of Microelectromechanical Systems (MEMS) resona-

tors is investigated. Local bifurcations of both the single resonator and the coupled

system are analyzed in the design of the coupled system. We discuss the behavior of the

coupled system according to the change in the coupling parameter and the excitation

numerical simulations and studied analytically through perturbation methods. The

coupled system shows oscillatory behavior without the excitation force. In the presence

of the excitation force the coupled system exhibits complex behavior which can be

beneficial in the design of the coupled system as a sensor.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

With the maturation and advancement of the MEMS technology, MEMS devices are being substituted for crystal oscillators inthe design of the control systems of the engine and are being utilized in the design of the sensors such as gyroscopes in theautomobiles [1]. In wireless communications MEMS devices are being used as high quality factor (Q) oscillator references in kHzrange, as bandpass filters, and as resonators in GHz range [1,2]. Depending on the method of actuation and the design, MEMSdevices can exhibit interesting nonlinear dynamics such as hard-spring effect, soft-spring effect, and parametric resonance [3].When two nonlinear oscillators are coupled together and excited by an external force, they exhibit in-phase oscillations withstrong coupling and out-of-phase oscillations with weak coupling [4]. In-phase oscillations are characterized by a phase differenceof zero and out-of-phase oscillations are characterized by a phase difference of p. In a network of weakly coupled MEMSresonators with symmetric coupling, this behavior can be used for pattern recognition such that the resonators can converge to aphase-locked pattern with identical frequencies and nonidentical phases [5]. The focus of our research is on phase-locked orfrequency-locked states produced by nonlinear MEMS resonators when they are coupled by discrete electronics in a ringformation. Complex dynamics in a ring of coupled nonlinear oscillators and gyroscopes have been investigated in [6–8]. The studypresented in this paper offers a synthesis of design and analysis of specific MEMS devices in the coupled system. A survey ofMEMS literature indicates that a MEMS device has not been shown to oscillate without an excitation force or without placing it ina closed feedback loop. We show that as a part of a coupled system, the MEMS resonator can exhibit oscillations at a specific value

All rights reserved.

), hikihara@kuee.kyoto-u.ac.jp (T. Hikihara), huykhanhvu@yahoo.com (H. Vu),

ar.navy.mil (V. In), patrick.longhini@navy.mil (P. Longhini).

S. Naik et al. / Journal of Sound and Vibration 331 (2012) 1127–11421128

of the coupling strength without an excitation force. A nonlinear autonomous system can be synchronized to an excitation forcewhen the excitation frequency is detuned [7]. In the presence of the excitation force with low amplitude, the coupled system inthis study shows multiple bifurcations depending on the interaction between damping constant, excitation amplitude andexcitation frequency. At a higher value of the excitation amplitude, the coupled system shows phase-locked behavior and fullentrainment to the excitation force. The coupling parameter and the excitation frequency also play a major role in which type ofbehavior is exhibited; thus the coupled system can be used as a sensor depending on the target application. The paper isorganized as follows. First in Section 2, the design process of a single resonator is described and its nonlinear resonance property isdiscussed. Next a coupled system with unidirectional coupling topology is introduced in Section 3. An autonomous systemconsisting of three resonators is studied numerically and analytically through perturbation methods in Section 4. Thesynchronization properties of the coupled system in the presence of excitation force are discussed in Section 5. Finally,application areas based on the simulation results are briefly discussed in Section 6 and conclusions are drawn in Section 7.

2. Design of single resonator

Fig. 1 illustrates a laterally driven comb-drive resonator that was fabricated in SOIMUMPs process. The resonatorconsists of a perforated mass which is suspended by four folded beam pairs. The folded beam pairs are attached to thetrusses. The device is designed to be symmetric in x- and y-axes to provide stable oscillations. In order to actuate thedevice, an ac voltage with dc voltage is applied between the movable comb fingers and the fixed fingers (either left or rightelectrode in Fig. 1). The remaining electrodes are grounded and as a result time-varying electrostatic force is generated.Consequently the structure moves in the direction of the force because of the compliance in the suspension beams in thatdirection. The folded beams exert the restoring force in the opposite direction to the applied force which then sustains themotion and ultimately creates maximum displacement at the resonant frequency. Typically the motional current, which isproportional to the change in the capacitance between the comb fingers and the excitation voltage, is measured. It isanticipated that the vibration in this type of resonator will be in the x direction due to the following design features: foldedbeams and thickness of the structure. These features reduce axial stress, restrict out-of-plane movement, and thereforeminimize unstable and unwanted vibrations in the other axes [9].

The devices were fabricated in a SOIMUMPs process. SOIMUMPs is a simple silicon-on-insulator (SOI) patterning andetching process offered by Memscap, Inc. Fig. 2 shows the layers used in the SOIMUMPs process. The end result is 25 mmthick doped silicon as the structural layer which is patterned and etched on 400 mm thick silicon substrate with 2 mm thickoxide as the insulating layer [10]. The substrate underneath the structure is completely removed by back-etching whichreduces parasitic capacitances and damping. The Pad Metal layer is used for the bond-pads for excitation and detection.Table 1 shows the important dimensions of the resonator.

2.1. Electrostatic force

For the device shown in Fig. 1, the dynamic capacitance between the movable fingers and the fixed fingers on eitherside can be written as

CðxÞ ¼2NE0ðsþxÞTth

g, (1)

Fig. 1. Fabricated resonator and pertinent dimensions.

Fig. 2. Layers used in the SOIMUMPs process [10].

Table 1Important dimensions of the device shown in Fig. 1.

Part description Designed value

Structure thickness (z-dir) 25 mm

Number of movable fingers (one side) 39

Gap between movable and fixed finger 3 mm

Overlap between movable and fixed finger 100 mm

Outer beam: width� length 5 mm� 306 mm

Inner beam: width� length 5 mm� 331 mm

Mass width� length 175 mm� 570 mm

S. Naik et al. / Journal of Sound and Vibration 331 (2012) 1127–1142 1129

where N is the total number of movable fingers, E0 the permittivity of the free space, s the overlap between the fixed andmovable fingers, x the displacement in x direction, Tth the thickness of the structural layer, and g the gap between the fixedand movable fingers on one side.

Using Eq. (1) and the principles of electrostatic actuation based on co-energy, the electrostatic force in laterally drivencomb-drive can be derived as

FðxÞ ¼1

2

qCðxÞ

qxV2, (2)

where V is the excitation voltage. After taking the derivative of C(x) with respect to x, the electrostatic force is simply

Fe ¼NE0TthV2

g: (3)

Eq. (3) indicates that in a laterally driven comb-drive, the electrostatic force exerted on the mass in x direction does notdepend on displacement in x direction. Here we assume that there is no movement in y direction and hence there is nochange in gap that can offset the movement in x direction. Also note that the capacitance between the movable fingers andthe boundary of the cavity is very small and can be ignored because the pertinent distance in x direction bg (see Fig. 1).We also ignore the fringe field capacitance.

Let the excitation voltage V ¼ VdcþVac cosðotÞ, where Vdc denotes the bias voltage and Vac denotes the amplitude of acvoltage. Then Fe becomes

Fe ¼NE0Tth

gV2

dcþ1

2V2

acþ2VdcVac cosðotÞþ1

2V2

ac cosð2otÞ

� �: (4)

Eq. (4) contains a constant plus a sum of two harmonic functions.1 If VdcbVac then the second harmonic term can beneglected. Additionally the effect of bias can also be ignored as it simply tilts the potential well and changes theequilibrium point. Thus the electrostatic force expression can be simplified as sinusoidal excitation with fundamentalexcitation frequency for the purposes of simulation.

2.2. Model of the resonator

The following equation represents a nonlinear MEMS resonator with a cubic spring constant term [11]

m €xþc _xþk1xþk3x3 ¼ Ad cosðodtÞ, (5)

1 Note that if a push–pull or differential excitation scheme is used then the constant dc terms cancel out completely because

Fe ¼ ð1=2ÞðqC=qxÞðV2þ�V2

�Þ ¼ 2ðqC=qxÞðVdcVac cosðotÞÞ.

S. Naik et al. / Journal of Sound and Vibration 331 (2012) 1127–11421130

where m denotes the mass, c is the damping constant to account for various mechanical and electronics losses. k1 and k3

correspond to the linear and nonlinear mechanical spring constants, respectively. Here od is the angular excitationfrequency and Ad is the excitation amplitude.

2.2.1. Nonlinearity

Large force (mainly due to dc bias) can create large deformation if deformation 40:15L approximately, where L is thelength of the beam [12]. Large deformation in the folded beam pair with identical lengths can cause hardening of thebeams as was observed in [13]. The folded beam pairs in Fig. 1 consist of fixed-guided beams. When the force is applied onthese beams, the energy potential is stored in the form of elastic deformation energy in the entire beam span which resultsin the linear elastic restoring force that is governed by Hooke’s law and additional reaction forces due to axial loading. As aresult the beams harden when they are driven by large force and this can lead to nonlinear restoring force.

For the design shown in Fig. 1, the mismatch between the lengths of the outer beam and inner beam within a given pairis yet an another cause for the hardening of the beams. Here the inner beams (IB) are designed to be slightly longer thanthe outer beams (OB). This creates a ratio between the inner beams and outer beams and introduces an asymmetry. WithLOB ¼ 306 mm and LIB ¼ 336 mm, kOB=kIB ¼ L3

IB=L3OB ¼ 1:32 (in x direction). As a result, the expansion/contraction experienced

by the outer beam is not fully matched by the contraction/expansion experienced by the inner beam. Here the outer beamswould harden more than the inner beams for the same range of displacement. Large vibrations in devices such as the oneshown in Fig. 1 can cause hard-spring effect at high dc bias as was shown in [14].

Both the soft-spring effect and the hard-spring effect in electrostatically excited devices such as fixed–fixed beamresonator and comb-drive resonator have been reported in MEMS research [11,13,15]. Incidentally the nonlinearity ischaracterized so that the device can be designed and operated with linear resonance [13] or the layout of the resonator canbe optimized [11]. Other research focuses phenomenologically on MEMS device with nonlinear resonance that can showM-cycles, period-doubling bifurcations and a route to chaos [3]. In this study the nonlinear behavior of an individualdevice is considered for the investigation of synchronized behavior in the coupled system.

The following equation represents the MEMS resonator with nonlinearity in dimensionless form by substituting t¼o0t

in Eq. (5):

x00 þdx0 þxþbx3 ¼ E cosðotÞ, (6)

where (0) and (00) represent first and second order derivatives with respect to the normalized time t. o0 denotes the naturalfrequency used in the normalization. Then it follows that d¼ c=ðmo0Þ, b¼ k3=ðmo2

0Þ, E¼ Ad=ðmo20Þ, and o¼od=o0.

2.2.2. Parameter values

In this section, the estimation of parameter values in Eq. (6) is presented.The value of the mass m is the least affected parameter during MEMS fabrication [9]. This value is calculated from

m¼ Ahr¼ 5:92� 10�9 kg, where A is the total area obtained from the layout, h is the structural thickness, and r is thedensity of silicon (about 2.33�103 kg/m3). From the spring design equations for the folded fixed-guided beams, the springconstant k1 is approximated as 30.97 N/m. This value is verified in Comsol by plotting an applied force versus displacementcurve for a range of applied force magnitudes and by estimating the slope. Using these values the natural frequency of thedevice is approximated as 11 512 Hz from f 0 ¼o0=2p, where o0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffik1=m

p. The value of f0 is also verified by running a

modal analysis and a frequency sweep analysis in Comsol. Fig. 3 shows first three modes that were simulated in Comsol.It is anticipated that the device is operated in the first mode. The truss reduces motion in the y-axis in this mode. Analysisof the other modes reveals that the frequencies corresponding to the other modes are higher than the fundamentalfrequency. In thin-film MEMS processes, the substrate underneath the structure is one of the major sources of energy lossdue to the effect known as squeeze-film damping [9]. As mentioned before, the substrate under the spring–mass structureis completely removed in the SOIMUMPs process. This etching step lowers the energy loss in the structure. Therefore it isanticipated that the loss of energy and lowering of the Q factor would be mostly due to the slide-film damping between thecomb fingers [9], the parasitic capacitances of the bond-pads, and the electronics. Additionally the energy loss can alsooccur due to the instability of the structure while being oscillated, viscous damping, and a finite volume of the anchor.Moreover the damping effect can be very high at atmospheric pressure (Q o100). After considering all of the above issues,the quality factor Q of the resonator is estimated as 282. With this value of Q, the resonator can be categorized between theresonators with high energy loss in the air and the resonators with high Q in the vacuum [13]. From Q ¼o0m=c, thedamping constant c is estimated to be 1.5�106 N s/m. For simulation purposes, the nonlinear spring constant k3 isestimated as 100 N/m3. Based on the above values, d is estimated as 0.0035, and b is estimated as 3.23 in Eq. (6). Afterfixing all other parameters, the variable parameters are E and o in Eq. (6).

2.3. Frequency response of single resonator

Amplitude (r) and phase (y) are obtained by using method of averaging on Eq. (6) as

r0 ¼1

2o ½�dor�E sin y�, (7a)

Fig. 3. Modal analysis in Comsol: (a) first mode at 11 511.95 Hz, (b) second mode at 67 725.23 Hz, and (c) third mode at 70 016.02 Hz. The bar on far-

right represents normalized values with most positive value on top depicting the maximum value of displacement for a given mode. Size and number of

the arrows are related to the values of displacement; maximum displacement of the mass in x-dir occurs in the first mode.

Fig. 4. Frequency response curves for E¼ 0:001 with up-sweep and down-sweep of frequency: (a) amplitude response with b¼ 0, b¼ 3:23, and b¼ 5,

(b) phase response with b¼ 0, b¼ 3:23, and b¼ 5. Dots represent stable points and circles represent unstable points.

S. Naik et al. / Journal of Sound and Vibration 331 (2012) 1127–1142 1131

y0r¼1

2o�ðo2�1Þrþ

3b4br3�E cos y

� �: (7b)

The exact solutions were numerically verified by using 4th-order Runge–Kutta method combined with Newton–Raphson method [16]. Fig. 4 shows the amplitude response and the phase response curves for E¼ 0:001 as o is swept.As expected linear resonance appears with b¼ 0 and the resonator shows hard-spring effect with b¼ 3:23 which results inthe hysteresis during up-sweep and down-sweep of the frequency. The width of the hysteretic region is defined by theend-points during up-sweep and down-sweep where the jump phenomena occur. The end-points depend on excitationamplitude, nonlinear spring constant, and damping constant. For example, the amplitude and the width of the hystereticregion increase significantly at E¼ 0:01, as was observed in the simulation. Similarly the curves bend to the right afterincreasing b to 5 and the usable bandwidth of the resonator is increased as shown in Fig. 4. Control of the width ofhysteresis is an important design feature as this characteristic is crucial in understanding the synchronized behavior ofcoupled resonators. Additionally it is worth noting that the value of the damping constant determines the tip of theamplitude response curve during the up-sweep.

The single resonator in Eq. (6) has a single well potential function. Note that Eq. (6) can be analyzed in two ways: theunforced system in which E¼ 0 and the forced system E40. In the unforced system, i.e. when E¼ 0, x¼0 is indeed anequilibrium. In the forced system, x¼0 is no longer an equilibrium. Instead a new set of equilibrium points exists and theycan be found by setting the right-hand-side of Eq. (7) to zero and solving for r and y. Finding that new set of equilibrium

S. Naik et al. / Journal of Sound and Vibration 331 (2012) 1127–11421132

points of the forced system can be, however, a daunting task. In fact, calculations of these equilibria lead to a generic formof the forced Stuart–Landau equation, which has a long history, see e.g. Appleton [17] and van der Pol [18,19]. A completepicture of the equilibria was not established until recently by Holmes and Rand [20] and Argyris [21]. In those works thestationary solutions are found by setting _r ¼ 0 and then solving for r from a 61 polynomial.

Here one stable periodic cycle appears at lower excitation frequencies. At a frequency value within the hystereticregion, two stable periodic cycles and an unstable periodic cycle exist indicating that the resonator has gone through asaddle-node (SN) bifurcation [22]. Outside the hysteretic region, one periodic cycle appears at higher frequencies.

3. Unidirectionally coupled MEMS resonators

In this section the single nonlinear resonator shown in Fig. 1 is employed in a system of coupled resonators in ringformation. The ring consists of an identical set of MEMS devices.

3.1. Coupling topology

Two types of coupling topologies are possible for the coupled system in ring formation: (a) diffusive and (b) direct [4].Diffusive coupling is defined by the coupling related to the difference between the displacement variables. For example,this type of coupling occurs when a beam is placed between the resonators. In the case of direct coupling, only theprevious element (or the next element if direction of coupling is reversed) produces the coupling term for a given elementwithin the coupled system as shown in Fig. 5. From design perspective, it is important for the coupling parameter to betunable as it provides more flexibility. As an example, this type of coupling can be implemented by placing current-to-voltage converters, op-amp gain stage, and op-amp buffers between the resonators. The gain of an amplifier with largegain-bandwidth product can be easily tuned so that the total coupling strength between the resonating elements ischanged linearly. Given that the buffers have a high input impedance and low output impedance, the flow of the motionalcurrent is always forward, i.e. from a given resonating element to the next resonating element. Consequently thisimplementation creates direct, linear, and unidirectional coupling scheme between the individual elements. Note that thiscoupling topology and its proposed implementation simplify the problem of the coupled system significantly.

Now the dynamics of a single resonator in the coupled system with direct coupling from the neighboring element canbe described as

xj00 þdx0jþxjþbx3

j ¼ E cosðotÞþljkxk, (8)

where xj denotes the displacement of a single resonator in the coupled system and ljk is the coupling strength with k

denoting the index of the previous element as is shown in Fig. 5. Here j¼(1,2,3) corresponds to k¼(3,1,2). To reduce thecomplexity and the number of parameters of the coupled system, it is assumed that (a) the coupling strength is identicalbetween the elements, i.e. l21 ¼ l32 ¼ l13 ¼ l, (b) the number of elements¼3, and (c) all other parameters are identical.The parameter space now contains three tunable parameters: coupling strength l, excitation amplitude E and excitationfrequency o.

The system shown in Eq. (8) has a rotational symmetry described by cyclic group Z3 with permutationðx1,x2,x3Þ/ðx3,x2,x1Þ [23]. The change in the coupling strength of rotationally symmetric system results in the solutionsthat have spatio-temporal symmetry [6]. Group theory combined with equivalent Hopf bifurcation theory can describefour possible states of phase-locking behavior in a ring of three coupled oscillators: (a) completely synchronized state,(b) one-third of a cycle out-of-phase with each other, (c) two synchronized to each other with third resonator withunrelated phase, or (d) two out of synchrony and the other at twice the frequency [24]. These states describe the genericbehavior of the coupled system. As will be shown, it is not possible to observe all four states for the coupled systempresented in this study.

4. Autonomous system (e¼0)

Here the coupled system without any excitation force is considered. Fig. 6 shows one parameter bifurcation diagram forx1 as l is varied. As shown in Fig. 6, when the coupling strength l is swept from right to left, the equilibrium point

Fig. 5. Unidirectional coupling scheme in a ring of coupled resonators.

Fig. 6. One parameter bifurcation diagram for x1 with l as the bifurcation parameter obtained numerically by integrating Eq. (8): Hopf bifurcation (HB)

occurs at a critical value of l¼�4:04� 10�3. Periodic oscillations continue until another bifurcation point at l¼�4:15� 10�3, where quasiperiodic

oscillations appear through Neimark–Sacker (NS) bifurcation.

S. Naik et al. / Journal of Sound and Vibration 331 (2012) 1127–1142 1133

x1¼x2¼x3¼0 loses its stability by going through a supercritical Hopf Bifurcation (HB) at a critical value of lc ¼�4:04�10�3 [22]. Another bifurcation point is marked as Neimark–Sacker (NS) where quasiperiodic cycles appear. The abovesimulation was performed by integrating Eq. (8) using 4th-order Runge–Kutta method.

4.1. Perturbation analysis

We now seek analytic expressions for the bifurcation points, HB and NS shown in the one-parameter bifurcationdiagram of Fig. 6, through perturbation methods. For convenience, we rewrite the governing equations of the coupledsystem (with E¼ 0) in the following form:

m €xjþc _xjþkxjþmx3j ¼ lxjþ1, (9)

where j¼ 1;2,3 mod 3. Rescaling time by t¼w0t, where w0 ¼ffiffiffiffiffiffiffiffiffiffik=m

p, we cast the original equation of motion in

dimensionless form

xj00 þecw0x0jþxjþemx3

j ¼ elxjþ1, (10)

where e¼ 1=ðmw20Þ, and ðÞ0 denotes differentiation with respect to t. In what follows we assume that mass m and natural

frequency w0 satisfy 1=ðmw20Þ51. This is equivalent to a resonator with a very large linear mechanical spring constant

kb1. Thus e51 and we then seek an asymptotic expansion for xj in powers of e as

xjðx,ZÞ ¼ x0jðx,ZÞþex1jðx,ZÞþe2x2jðx,ZÞþ � � � , (11)

where x¼ ðo0þe2o2þe3o3þ � � �Þt denotes the fast scale while Z¼ et denotes the slow scale. In order to introduce thesetwo-time scales into Eq. (9), we need expressions for the first and second derivatives of x and y with respect to t, which weobtain through the chain rule:

dxj

dt ¼orqxj

qxþe

qxj

qZ ,

d2xj

dt2¼o2

r

q2xj

qx2þ2eor

q2xj

qxqZ þe2 q

2xj

qZ2, (12)

where or ¼o0þe2o2þe3o3þ � � � and o0 ¼ 1. We use f and a to denote the phase difference and ratio of amplitudesbetween the vibrations of the jth and (jþ1)th elements, so that

xjþ1 ¼ axj expð�ifÞ: (13)

In addition to the asymptotic expansion for xj, see Eq. (11), we also write f and a in powers of e:f¼f0þef1þ � � � , (14a)

a¼ a0þea1þ � � � , (14b)

S. Naik et al. / Journal of Sound and Vibration 331 (2012) 1127–11421134

where, without loss of generality, we assume f040 and a040 while the other coefficients of the higher order powers of e,i.e. fj and aj, j40, can be either negative or positive as they act as a correction of the approximation. Using Eq. (14a), thephase term expð�ifÞ in Eq. (13) can also be written in a truncated form:

expð�ifÞ ¼ expð�if0Þ�ief1expð�if0Þþh:o:t:, (15)

where h.o.t. means higher order terms. Substituting Eqs. (11), (14a) and (15) into Eq. (13) we get

xjþ1 ¼ a0x0j expð�if0Þþeð�ia0x0jf1þa0x1jþa1x0jÞexpð�if0Þ: (16)

Direct comparison of Eq. (16) with Eq. (11) leads to

x0,jþ1 ¼ a0x0j expð�if0Þ, (17a)

x1,jþ1 ¼ ð�ia0x0jf1þa0x1jþa1x0jÞexpð�if0Þ: (17b)

Substituting Eqs. (11)–(17) into the governing equation (10) yields, after collecting equal powers of e, a set of partialdifferential equations (PDE) associated with each order term:

Order e0:

q2x0j

qx2þx0j ¼ 0: (18)

Order e:

q2x1j

qx2þx1jþ2

q2x0j

qxqZþcw0

qx0j

qxþmx3

0j ¼ lx0,jþ1: (19)

Order e2:

q2x2j

qx2þx2jþ2o2

q2x0j

qx2þ2

q2x1j

qxqZ þq2x0j

qx2þcw0

qx1j

qxþ

qx0j

qZ

� �þ3mx2

0jx1j ¼ lx1,jþ1: (20)

Solving Eqs. (18)–(20) we arrive at closed form solutions for x0j, x1j and, eventually, to the desired approximation for xj.We outline next only the main results. We can assume a¼ 1 since the Hopf Bifurcation leads to limit cycle oscillations ofidentical amplitude among all resonators. Numerical simulations show that these oscillations form a traveling wavepattern so we assume f¼ 2p=3. Eliminating secular terms from the PDEs (18)–(20) lead to an underdetermined algebraicsystem of equations for a and f. Among the infinite set of solutions, we considered two cases: one where a0 depends on lwhile f0 is held constant and one where f0 depends on l while a0 is now constant. In the former case, the solutions for a0

and a1 are

a0ðlÞ ¼�2cw0

ffiffiffi3p

3l,

a1ðlÞ ¼1

e1þ

2cw0

ffiffiffi3p

3l

" #: (21)

The argument in these two equations indicates that a0 and a1 vary as functions of the coupling strength. Eliminatingsecular terms, and using Eqs. (21), we eventually arrive at the desired solution for the displacement in the resonator

xj,a ¼ a0j,a cos os,at�2p3ðj�1Þ

� �þ

e32

ma30j,a cosð3os,atÞþOðe2Þ, (22)

where

a0j,a ¼2

3ecw0m

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiecw0m½ec2w2

0

ffiffiffi3p�6cw0�3

ffiffiffi3p

l�q

,

os,a ¼ 1þeo1,aþe2o2,a:

o1,a ¼38 ma2

0j,a�ffiffi3p

6 cw0,

o2,a ¼1

768e ½192ffiffiffi3p

ecw0ma20j,a�244ec2w2

0þ768lþ512ffiffiffi3p

cw0�45em2a40j,a�:

From Eq. (22) it can be seen that the Hopf bifurcation point corresponds to the critical value of l where the amplitude ofthe oscillations is no longer zero. Thus setting a0j,a equal to zero, yields

lHBa ¼ffiffi3p

9 cw0½ecw0

ffiffiffi3p�6�: (23)

S. Naik et al. / Journal of Sound and Vibration 331 (2012) 1127–1142 1135

The Neimark–Sacker bifurcation point corresponds to the point where a0 ¼ a1, which yields

lNSa ¼�2ffiffi3p

3 cw0ð1þeÞ: (24)

In the case where f0 and f1 depend on l we get

f0ðlÞ ¼ pþsin�1 cw0

a0l

� �,

f1ðlÞ ¼�1

ep3þsin�1 cw0

a0l

� �� �: (25)

Again, eliminating secular terms and using Eqs. (25), we arrive at the solution for the displacement in the resonator

xj,a ¼ a0j,f cos os,ft�2p3ðj�1Þ

� �þ

e32

ma30j,f cosð3os,ftÞþOðe2Þ, (26)

where

a0j,f ¼2

3ecw0m�ecw0ml

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�

c2w20�l

2

l2

s3ecw0þ2pþ6 sin�1 cw0

l

� �h i8<:

9=;

1=2

,

os,f ¼ 1þeo1,fþe2o2,f,

o1,f ¼3

8ma2

0j,fþ1

2l

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

c2w20

l2

s,

o2,f ¼1

768ecw0�45ecw0m2a4

0j,f�96ec3w30þ96ecw0l

2þ128pl2

þ384 sin�1 cw0

l

� �l2

h i:

In a similar manner, the Hopf bifurcation point is detected by setting a0j,f equal to zero, which yields

lHBf ¼�cw0

sinecw0

2þp3

� �, (27)

while the Neimark–Sacker bifurcation point is obtained by setting 9sin f09¼ 9sin f19, which yields

lNSf ¼�cw0

a0 sinp

3ðeþ1Þ

� �: (28)

A comparison of Eq. (23) with Eq. (27) shows that one underestimates the critical coupling strength that leads to a Hopfbifurcation while the other overestimates the same value. Then an average between these two equations acts as acorrection term and leads to a better approximation. Similarly, an average between Eq. (24) and Eq. (28) yields a betterapproximation for the critical coupling strength associated with the Neimark–Sacker bifurcation points. In summary we get

lHB ¼1

2

ffiffiffi3p

9cw0½ecw0

ffiffiffi3p�6��

cw0

sinecw0

2þp3

� �264

375, (29a)

lNS ¼�1

2

2ffiffiffi3p

3cw0ð1þeÞþ

cw0

a0 sinp

3ðeþ1Þ

� �2664

3775: (29b)

Fig. 7 shows a one-parameter bifurcation diagram of the average amplitude response as a function of coupling strength l. Thediagram was obtained numerically with the aid of the continuation software package XPPAUT [25] and analytically by plottinga0j,avg ¼ ða0j,aþa0j,fÞ=2 as a function of l. Observe that the analytical bifurcation diagram matches very well with the numericalone and also with the diagram shown in Fig. 6.

Next we focus on the stability analysis and the rest of the simulations by referring to Eq. (8).

4.2. Stability of the equilibrium

Stability of the synchronous equilibrium was verified by plotting asymptotic continuous spectrum (ACS) curves alongwith discrete eigenvalues for each resonator [6]. Fig. 8a shows one set of the complex conjugate eigenvalues for different lvalues. The ACS curves shown in Fig. 8a represent a continuous set of eigenvalues for each l with discrete eigenvalues ofeach resonator shown as big dots on these curves. For l4lc all of the eigenvalues stay within the left-hand plane.

Fig. 7. One-parameter bifurcation diagrams obtained numerically with the aid of the continuation software XPPAUT and analytically through the average

amplitude response a0j,avg ðlÞ. Note that le ¼ el in order to compare directly the results using Eq. (8) with those from Eq. (10). Also see Fig. 6.

Fig. 8. Stability of the equilibrium: (a) asymptotic continuous spectrum (ACS) curves for l¼ l1 ¼�1� 10�3, l¼ l2 ¼�4:08� 10�3, and

l¼ l3 ¼�5� 10�3. Inset shows the zoomed view of one of the eigenvalues for l2 and l3. (b) Real part of eigenvalues vs. l with lc as marked.

S. Naik et al. / Journal of Sound and Vibration 331 (2012) 1127–11421136

For lolc , one of the eigenvalues crosses into the right-hand plane. As shown in Fig. 8b, the critical value of lc can also beconfirmed by evaluating the real part of eigenvalues while sweeping l values.

Periodic rotating waves appear at a critical value of l for which the frequency of the oscillation was observed as 1.008.Fig. 9a shows the time-series of the coupled system at l¼�4:08� 10�3 which is past the critical value. A single oscillationfrequency appears in the power spectrum as shown in Fig. 9b. As seen from Fig. 9c, the phase-space encloses a circle. Thephase difference is also evident from the Poincare section shown in Fig. 9d. Stability of the periodic solutions is done byevaluating eigenvalues of the system on the linearized Poincare map. Here two complex conjugate pairs stay within theunit-circle while the complex conjugate pair for the third solution stays around 1 (see Section 4.3 for the method used forstability check). Both the frequency and the amplitude of the oscillations increase by decreasing the bifurcation parameteras shown in Figs. 6 and 7. These periodic cycles are stable up to l¼�4:15� 10�3, at which the system goes through asecondary Hopf bifurcation. At this point one of the complex conjugate pair exits the unit-circle indicating that systemgoes through Neimark–Sacker (NS) bifurcation as marked in Figs. 6 and 7 [22]. Fig. 10 shows the state of the system whenl is changed to �4.28�10�3. As shown in Fig. 10a, the original phase-locked state between the three oscillators is nowdestroyed and the system shows amplitude-modulated periodic cycles. Multiple frequency components around theoriginal oscillation frequency appear in the power spectrum as shown in Fig. 10b. Note that the third harmonic

Fig. 9. Simulation of coupled system for l¼�4:08� 10�3 and E¼ 0: (a) time-series for x1, x2, x3, (b) power spectrum for x1, (c) phase-space for x1, x2, x3,

and (d) Poincare section with ( _x1 , x1), ( _x2 , x2), and ( _x3 , x3).

S. Naik et al. / Journal of Sound and Vibration 331 (2012) 1127–1142 1137

components also begin to appear for l¼�4:28� 10�3. The phase-space now evolves into a torus as seen in Fig. 10c.Individual oscillators follow a closed loop on the Poincare section as seen in Fig. 10d. The behavior shown for negative lwas also observed for a positive change in l with the critical value lc ¼ 4:04� 10�3.

Note that nonidentical and random initial conditions were used in all the simulations. Indeed this is similar to theactual devices in which noise in the electronics and variation in the fabrication process make the initial positions of thethree resonators nonidentical. Identical initial conditions can cause oscillation death in this system because suchconditions are part of the invariant manifolds where equilibrium points are located. Even if these manifolds are unstable,initial conditions that lie within them will remain there at all times due to their invariant properties. Calculations andvisualizations of basins of attractions and invariant manifolds is beyond the scope of this paper.

4.3. Stability of the periodic solution

Floquet multipliers are used to determine the stability of periodic solution for both the autonomous system and thenonautonomous system (see Section 5). A detailed discussion can be found in [22]. Floquet multipliers are the eigenvaluesof the monodromy matrix. If the eigenvalues of the monodromy matrix are located inside the unit-circle, i.e. if the modulusof the complex eigenvalue is less than 1, then the periodic orbit is considered to be stable. By examining the values of theFloquet multipliers (r) corresponding to the unstable solution, the bifurcation can be classified as: (a) saddle-node orpitchfork if 9r9¼ 1, (b) Neimark–Sacker if ImðrÞa0, where Im denotes imaginary part, and (c) period-doubling if 9r9¼�1[22,26].

4.4. Discussion

As shown in Fig. 11a, for l¼�4:08� 10�3 one set of eigenvalues stays at 1 and corresponding eigenvectors are tangentto the periodic orbit through the Poincare section, indicating neutrally stable perturbation [22]. The other Floquetmultipliers stay inside the unit-circle indicating stable periodic solution. As shown in Fig. 11b for l¼�4:28� 10�3, onepair of Floquet multipliers exits the unit circle indicating unstable periodic solution. It is interesting to note that an anothereigenvalue is located outside the unit-circle on the real axis. While decreasing l past the NS point, which is marked inFig. 6, the system has gone through a saddle-node (or pitchfork) bifurcation together with Neimark–Sacker bifurcation.

Fig. 10. Simulation of coupled system for l¼�4:28� 10�3 and E¼ 0: (a) time-series for x1, x2, x3, (b) power spectrum for x1 and the zoomed view around

fundamental frequency and its third harmonic, (c) phase-space for x1, x2, x3, and (d) Poincare section with ( _x1 , x1), ( _x2 , x2), and ( _x3 , x3).

Fig. 11. Floquet multipliers of the autonomous system for two values of l: (a) l¼�4:08� 10�3; one complex conjugate pair stays at 1 making the

system neutrally stable. The rest of the multipliers are located inside the unit-circle. (b) l¼�4:28� 10�3; the complex conjugate pair is located outside

the unit-circle indicating that the system has gone through Neimark–Sacker (NS) bifurcation. The third eigenvalue outside the unit-circle indicates that

system has gone through a saddle-node (or pitchfork) bifurcation as well.

S. Naik et al. / Journal of Sound and Vibration 331 (2012) 1127–11421138

After going through the Hopf bifurcation, all periodic solutions of a single resonator within the coupled system maintain afixed phase difference with the neighboring element. Generally in a ring of n-coupled elements (unidirectionally orbidirectionally coupled), the system exhibits 2p=n phase difference between neighboring elements for odd n [7,27].Similarly in this system with n¼3, 2p=3 phase difference appears for all stable periodic cycles. Past a specific value of l, thesymmetry between the three oscillators in a phase-locked state breaks and the system goes through a secondary Hopfbifurcation leading to a torus without the spatio-temporal symmetry as seen in Fig. 10c. Decreasing the couplingparameter beyond this value causes the system to undergo global symmetry-breaking bifurcations in a sequence whichresults in multiple tori in the phase space. Poincare section generated for the values of l in the multiple tori region shows aclosed ring containing the three individual oscillator paths shown in Fig. 10d. The simulations show that these bifurcationslead to multiple tori with each bifurcation adding another fundamental frequency till the trajectories of three elements

Fig. 12. Regions of synchronization for l¼�4:08� 10�3 around o¼ 1 and o¼ 3. Here circles represent numerically obtained boundary and the solid

line is the fitted curve. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

S. Naik et al. / Journal of Sound and Vibration 331 (2012) 1127–1142 1139

envelope each other by surrounding the individual loops charted by each element. This suggests that the coupled systembecomes more unstable as more frequencies are added. Note that this change occurs in a narrow range of the couplingparameter.

5. Nonautonomous system (ea0)

With a nonzero E, the state of the coupled system changes drastically. As shown in Eq. (8), the excitation force isidentically applied to all three resonators. Fig. 12 shows the synchronization regions for l¼�4:08� 10�3 with detuningeffects around o¼ 1 and o¼ 3. Recall that o is the ratio of the excitation frequency and the natural frequency of thecoupled system which depends on the coupling strength. The simulations show that inside the Arnold tongue where o isaround 1, the individual resonators are fully entrained to the excitation frequency. For lower o values, 1:1 region showscurved boundary which indicates that higher order nonlinear terms are present as this region is mixed with thesubharmonic regions, e.g. 3:1 region. Albeit of having a narrow region around o¼ 3, the coupled system shows 2p=3phase-locked behavior similar to Fig. 9a. Here the frequencies of the individual resonator are locked at 1/3rd of theexcitation frequency. Outside the Arnold tongues, the coupled system shows almost periodic oscillations near theboundaries and quasiperiodic oscillations away from the boundaries. With large d (damping constant) and small E, 1:1sync region can show a closed curve above the area marked with red square in Fig. 12. For the value of d used in this study,no synchronization was observed in this area hence the open region. This region shows unstable responses due to thesimultaneous effects of damping, coupling strength, and the excitation force. As an example, for E¼ 0:001 and o¼ 1:05,the system loses its original synchronized state as seen in Fig. 13a and the interaction of the three oscillators results in therise of the third harmonic components as seen in Fig. 13b. Also note that amplitude level of the third harmonic continuesto rise with multiple spectral components for EZ0:001. Local stability analysis shows that one complex conjugate pairleaves the unit-circle indicating Neimark–Sacker bifurcation as seen in Fig. 14a. When E is increased beyond this value, thesystem first goes through a saddle-node (or pitchfork) bifurcation at E¼ 0:0031 before the stable periodic oscillations beginto appear at a higher value. These differences indicate a qualitative change in the behavior at low E. Fig. 14b shows Floquetmultipliers for o¼ 0:5 and o¼ 1:9 at E¼ 0:035 while sweeping o. The system goes through torus (NS) bifurcation at thesedetuned values of o.

For l¼�4:28� 10�3 the unstable system shown in Fig. 10 synchronizes in the presence of excitation force. Here thesystem shows many spectral components until full entrainment occurs at large E.

5.1. Discussion

From Fig. 10 we can see that for a specific value of the coupling strength, the coupled system with symmetry-breakingbifurcations shows quasiperiodic oscillations. It is possible for such a system in a priori oscillatory state to be synchronizedat an excitation frequency exactly 3 times the oscillation frequency [7,26]. This and the other types of behavior observed inthis system in the presence of the excitation force can be classified into three distinct states: (a) unstable, (b) fullyentrained to the excitation frequency, and (c) phase-locked and synchronized at a ratio of the excitation frequency.

After detuning excitation frequency at small E in the region shown as the red square in Fig. 12, the coupled system in apriori oscillatory state breaks the symmetry as shown in Fig. 13. The phase difference between individual elements and theexcitation force does not remain constant which makes the system unstable. As shown in Fig. 14a, while sweeping Eat o¼ 1:05 the system goes through an interesting series of bifurcations; from torus (NS) bifurcation to saddle-node

Fig. 13. Simulation of coupled system for l¼�4:08� 10�3, E¼ 0:001, and o¼ 1:05: (a) time-series for x1, x2, x3, (b) power spectrum for x1 and the zoomed view

around fundamental frequency and its third harmonic, (c) phase-space for x1, x2, x3, and (d) Poincare section with ( _x1 , x1), ( _x2 , x2), and ( _x3 , x3).

S. Naik et al. / Journal of Sound and Vibration 331 (2012) 1127–11421140

(or pitchfork) bifurcation which leads to almost-periodic oscillations. In this region damping can be the prominent cause ofsuch a behavior.

In particular the following behavior was observed while increasing E from 0.001 to 0.01 adiabatically in this region.At E¼ 0:001, the largest pair of the multipliers stays outside the unit-circle. This pair moves inside the unit-circle afterincreasing E. After increasing E from this value, the largest pair merges at the real axis and then they move in oppositedirection on the real axis as one of them eventually crosses the unit-circle. While increasing E from here the largest pairshows transitory behavior, e.g. moving inside the unit-circle and moving out of the unit-circle on the real axis. IncreasingE after this value makes the pair move inside the unit-circle. Similar behavior was observed for d¼ 0:01 with an appropriatechange in the other parameters. Therefore this behavior can be attributed to damping before verifying it experimentally. Theeffect of damping is overcome by increasing the excitation amplitude and this results in periodic oscillations which are fullysynchronized at the excitation frequency. An identical behavior occurs around 1:3 sync region at small E.

In all of the above states, the detuning between individual resonators raises the amplitude level of third harmonicfrequency components. Note that the system still maintains its original frequency of oscillations. As noted before,increasing the value of excitation amplitude causes the system to synchronize in 1:1 and 1:3 sync regions shown in Fig. 12.Here at higher excitation amplitude and at frequency outside the sync regions but close to the boundaries, almost periodicoscillations are observed. The synchronized behavior resembling state (c), which was described in the beginning of Section3, i.e. two resonators synchronizing to each other while the third resonator showing an unrelated phase, is observed whiledetuning the frequency near the boundaries. Detuning the frequency farther away from the boundaries causes the coupledsystem to go through symmetry breaking bifurcations leading to quasiperiodic oscillations. Local stability check for thefrequency values away from the boundary confirms this phenomena as is shown in Fig. 14b.

These simulations show how the coupled system reacts when the excitation amplitude is changed and when thefrequency of the forced system is detuned by the excitation frequency. The interplay between these two factors governshow the system in a priori oscillatory state can be synchronized for a given l. Excitation amplitude tends to bring thesystem closer to the synchronized state as it is an attractive factor while the detuning between the excitation frequencyand the frequency of the forced system acts as the repulsive factor.

Simulations were also performed in the region of parameter space where 9l9o9lc9. These simulations indicate that thesystem can start oscillations only in the presence of the excitation signal. Indeed this behavior is similar to the case ofuncoupled individual resonators.

Fig. 14. Floquet multipliers in two sweep directions around 1:1 sync region shown in Fig. 12: (a) E¼ 0:001,0:0031,0:0052 at o¼ 1:05. While sweeping Efrom low to high, at E¼ 0:001 the system goes through Neimark–Sacker (NS) bifurcation, at E¼ 0:0031 the system goes through saddle-node (SN)

bifurcation, and at E¼ 0:0052 the system shows almost-periodic oscillations. From this point as E is increased all the multipliers stay inside the unit-circle

and the system eventually shows periodic oscillations. (b) o¼ 0:5 and o¼ 1:9 at E¼ 0:035. As o is swept from the center of 1:1 sync region past the left

boundary and the right boundary, one complex-conjugate pair moves out of the unit-circle indicating NS bifurcation at o¼ 0:5 and at o¼ 1:9,

respectively.

S. Naik et al. / Journal of Sound and Vibration 331 (2012) 1127–1142 1141

5.1.1. Resonator mismatch

In this study individual MEMS resonators with identical natural frequencies were considered. It is possible forresonators on different chips or on the same chip to have nonidentical natural frequencies due to fabrication errors.Additionally the Q factors and the peak amplitude levels can be different. The effect of these variations can be posited asdeviation from an exact 2p=3 phase difference between two neighboring elements. In this case the coupling strengths needto be adjusted in order to induce self-excited mode in the coupled system. If the ratios of coupling strengths are notperfectly matched, the coupled system can exhibit in-phase vibrations as well as amplitude variation. Consequently thesynchronized behavior of the coupled system under excitation force may also change significantly. Simulations performedfor nonidentical natural frequencies show that the coupled system does synchronize in the presence of the excitationforce. The fully synchronized state of the coupled system is then similar to the injection-locking of MEMS oscillators withdifferent natural frequencies by an external force [15].

6. Recommended applications

There are many interesting application areas of the coupled system shown in this study.The coupled system can be used to detect mechanical excitation and its effect on region of synchronization can be

analyzed in a similar manner as the ac excitation once the experimental setup is fully functional. After investigating theeffect of mechanical excitation analytically and experimentally, the synchronized behavior can be explored for energyharvesting.

Around the boundaries of the sync region, the system is very sensitive and shows a rapid transition from unstable tostable vibrations. This property is advantageous to the coupled system so that it can be used in self-excited and forcedmode in the inertial sensors to increase sensitivity and robustness in the presence of noise. The application of the coupledsystem as inertial sensor is particularly attractive because the sync region is sensitive with respect to the couplingstrength. The coupled system of MEMS resonators can be employed in the drive axes implementation of the coupledgyroscope system [8].

An intriguing direction of the research is the implementation of the coupled system as a sensing element to detect anexcitation force with dependence on frequency and damping. For example, damping is the dominant factor for closed and/or wide sync regions at low excitation amplitude; this property of the system can be used for pressure sensing.

S. Naik et al. / Journal of Sound and Vibration 331 (2012) 1127–11421142

In the self-excited mode, the frequency range can be changed by varying the coupling strength and hence the coupledsystem can be used for frequency synthesis. This feature can be further developed as the MEMS implementation ofmultifrequency synthesis shown in [28].

7. Conclusions

In this paper we have discussed the design of a MEMS resonator with nonlinear resonance and simulated its behavior ina coupled system of resonators. It was shown by numerical and analytical methods that the coupled system can oscillatewithout an excitation signal at a specific value of the coupling strength and can also exhibit quasiperiodic cycles bychanging the coupling strength. Adding an external force to the coupled system in a priori oscillatory state changes thedynamics significantly. Phase-locked and frequency-locked states of the coupled system were also discussed. Themultitude of properties discussed above indicates that the present system of coupled nonlinear resonators can be usedas a sensor depending on the applications. The device shown in Fig. 1 is being individually characterized and as a part ofcoupled system in an experimental study.

Acknowledgments

This work was supported in part by Global Center of Excellence (GCOE) program at Kyoto University, by the Office ofNaval Research (Code 30), by ONR NREIP Internship Program, and by the SSC Pacific internal research funding (S&T)program. H.V. and A.P. were supported in part by National Science Foundation (NSF) Grants CMMI-0923803 and CMS-0625427. S.N. was partially supported by NSF Grant CMMI-0923803 to visit and conduct research with the group atSan Diego State Univ and SSC Pacific. S.N. and T.H. wish to thank Dr. Takashi Hisakado, Dr. Osamu Tabata, Dr. ToshiyukiTsuchiya, and Mr. Hiroyuki Tokusaki (Kyoto University) for helpful discussions. We also wish to thank Dr. Sam Kassegne(San Diego State University) for supporting our design efforts.

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