Parametrically Excited Electrostatic MEMS Cantilever Beam ... · Parametrically Excited...

Parametrically Excited Electrostatic MEMSCantilever Beam with Flexible Support

Mark Pallay, Shahrzad Towfighian∗

Department of Mechanical EngineeringBinghamton University4400 Vestal Parkway E.Binghamton, NY 13902.

AbstractParametric resonators that show large amplitude of vi-

bration are highly desired for sensing applications. In thispaper, a MEMS parametric resonator with a flexible sup-port that uses electrostatic fringe fields to achieve resonanceis introduced. The resonator shows a 50 percent increasein amplitude and a 50 percent decrease in threshold volt-age compared with a fixed support cantilever model. Theuse of electrostatic fringe fields eliminates the risk of pull-inand allows for high amplitudes of vibration. We studied theeffect of decreasing boundary stiffness on steady-state am-plitude and found that below a threshold chaotic behaviorcan occur, which was verified by the information dimensionof 0.59 and Poincare maps. Hence, to achieve a large am-plitude parametric resonator, the boundary stiffness shouldbe decreased but should not go below a threshold when thechaotic response will appear. The resonator described in thispaper uses a crab-leg spring attached to a cantilever beamto allow for both translation and rotation at the support. Thepresented study is useful in the design of mass sensors usingparametric resonance to achieve large amplitude and signalto noise ratio.

1 IntroductionMicro-electromechanical systems (MEMS) are used ex-

tensively in modern electrical, mechanical, and biologicaldevices. With new technology becoming both more com-plex and space efficient, MEMS are a necessity to develophigh performance electronics. MEMS resonators are a classof dynamic MEMS devices that utilize a vibrating beam orplate for mass and pressure sensing [1], gyroscopes/motiondetection [2], energy harvesting [3], and many other applica-tions [4, 5]. Their small size allows for extremely sensitive

∗Address all correspondence to this author at [email protected]

systems that consume minimal energy and have low fabrica-tion costs. With the desire for self powered resonators, sig-nificant effort has been made to further reduce energy con-sumption without sacrificing performance [3,5–7]. For manyelectrostatic MEMS resonators, optimization is aimed at in-creasing the amplitude of vibration while minimizing the ex-citation voltage. Large vibration amplitudes are desirable be-cause they result in a high signal to noise ratio for sensors,which increases sensitivity. A low excitation voltage reducesthe overall power consumption of the resonator, creating amore energy efficient sensor.

Parametric excitation (PE) has been used in MEMS res-onators [3, 6, 8–11] due to its low threshold voltage and highamplitude of vibration at frequencies far from the natural fre-quency of the system [12,13]. PE refers to excitation throughtime varying parameters of the system rather than through di-rect forcing in the direction of the desired motion [6]. Thisphenomenon was first noted by Faraday in the 1800s whilestudying the motion of fluid waves in closed cylinders [12].In the early 1990’s, Rugar and Grutter applied this principleto a MEMS cantilever resonator through a parametric ampli-fier and harmonic forcing [14]. Turner expanded this workby classifying the parametric instability regions/tongues in1998 [15], which has laid the groundwork for parametricallyexcited MEMS resonator designs.

The most common method of PE involves imposing atime varying force in one direction to amplify a vibration ina perpendicular direction. In MEMS sensors, this is achievedin a two degree-of-freedom (DOF) system where one direc-tion of vibration is dedicated to sensing, and a perpendiculardirection is used for actuation [2]. If the actuator is driven ata frequency within an instability tongue of the system, it candrive the sensing direction into resonance. Another way ofachieving parametric resonance (PR) in MEMS is to modu-late the effective stiffness of the resonator with time, usingpiezoelectrics, magnetism, or electrostatics [6]. In electro-static MEMS, the effective stiffness is dependent on boththe mechanical stiffness of the beam, and the electrostaticforces between the beam and electrode. If the stiffness is

VIB-16-1229 . Towfighian . 1

Journal of Vibration and Acoustics. Received April 21, 2016; Accepted manuscript posted October 12, 2016. doi:10.1115/1.4034954 Copyright (c) 2016 by ASME

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modulated in a way such that the restoring force of the beamis smaller when it deflects away from the equilibrium pointcompared to when it returns, the beam can show large oscil-lations (parametric resonance). The oscillations reach steadystate when the beam has deflected far enough that the me-chanical restoring force of the beam dominates and preventsthe beam from deflecting further at the applied voltage.

The effect of flexible supports, on the other hand, havebeen extensively studied for MEMS beams. In 2000, Ko-brinsky [16] developed an analytical model for a clamped-clamped electrostatic microbeam that takes into account sup-port compliance. This was used to help explain initial beamdeflections that had a significant effect on the dynamic re-sponse of the beam. Pakdemirli [17] studied the effect of aflexible boundary on the natural frequency and steady stateamplitude of several beam types. It was shown that flexi-ble supports may either increase or decrease both the natu-ral frequency and steady state amplitude, depending on thenature of the boundary conditions. Rinaldi [18] created amodel for a cantilever micro-beam with non-ideal supports.Comparisons between the ideal and non-ideal boundary forthe cantilever, regarding static tip deflection and natural fre-quency, were conducted. Clamped-Clamped arches withnon-ideal supports were investigated by Alkharabsheh andYounis [19, 20]. Further research of the dynamic effect ofnon-ideal boundary conditions for various types of beamshas been reported in [21–23].

A major obstacle for MEMS electrostatic resonators isthe pull-in instability that severely limits their range of mo-tion. This occurs when the electrostatic forces pulling elec-trodes together overcomes the mechanical forces separatingthem and the beam collapses to the electrode. For dynamicMEMS, such as resonators, this phenomena occurs at evenlower voltages (dynamic pull-in) than if the devices was op-erated statically. The pull-in voltage ultimately depends onthe geometry of the resonator, however in most cases thebeam can only reach from 33 to 50 percent of the gap dis-tance before pull-in occurs. Since electrostatic forcing is in-herently non-linear, the beam must be close to the electrodeto increase the magnitude of the force. This can yield pull-in voltages under 2 V for cantilever beams that are in closeproximity to the electrode [12].

Utlizing electrostatic fringe fields, Linzon et al. [6] in-vestigated a MEMS parametric resonator that does not sufferfrom pull-in instability. The system contained a long, thincantilever beam surrounded by an electrode at its free end.In their study, the electrostatic force acts as a restoring force,whose stiffness is a function of AC voltage and varies withtime. Varying the stiffness with time drives the beam intoparametric resonance. Large vibration amplitudes have beenachieved; however, because the fringe field generates a weakelectrostatic field, the threshold voltage necessary to induceparametric resonance is high.

The contribution of this paper is to reduce the thresholdvoltage of a fringe field MEMS paramteric resonator whilemaintaining a high vibration amplitude to increase the sig-nal to noise ratio. This is achieved by adding a flexiblesupport to the resonator as shown in Figure 1. Many res-

onators, including [6], use fixed supports. Fixed supports re-quire the beam to bend to deflect from the equilibrium point,which is a severe limitation on the vibration amplitude ofthe beam because of the non-linear stiffness at large defor-mations. The flexible support allows for rotation and trans-lation at the fixed boundary, yielding less overall resistanceto the deflection of the beam. This design can theoreticallyachieve higher amplitudes at lower voltages because it al-lows for a deflection without bending the beam. As bendingof the beam is what dominates the restoring force at highamplitudes, the less the beam bends, the smaller the restor-ing force. With a lower restoring force at any given deflec-tion, the beam can deflect to a larger amplitude under thesame magnitude of electrostatic force. Unlike previous stud-ies on non-ideal supports [19–23], which justified dynamicbehaviors by modeling the base with springs attached, wepurposely added a flexible support to increase vibration am-plitudes.

The flexible support is achieved by adding a crab-legspring at the fixed end of the cantilever. This configurationis chosen because it allows for easy control of both the trans-lational and rotational stiffness. This is used over a serpen-tine configuration (the typical MEMS torsional spring) be-cause the torsional stiffness of the serpentine spring is muchlower than translational stiffness, while a balance betweenthe two is needed to create large amplitudes (refer to section2.3). We demonstrate decreasing the boundary stiffness be-low a threshold results in a chaotic response, which shouldbe avoided if the resonator is used as a sensor.

The organization of this paper is as follows. Section2 outlines the derivation of the theoretical model with theadded flexible support, as well as the simulation results. InSection 3, a physical model and results are discussed. Thefinal conclusions based on the physical model’s results aregiven in Section 4.

2 Theoretical Model2.1 Equation of Motion

The governing equation of motion for the beam is ob-tained using Euler-Bernoulli beam theory. The dimensionsof the beam are given in Table 1. The beam material is sin-gle crystal silicon and assumed to be linearly elastic. Thismodel assumes only deflection in the z-direction, with no ax-ial/lateral displacement and no twisting of the beam. There-fore, the z-displacement of the beam is only dependent onthe position along the beam (x) and time (t). The dimension-alized equation of motion (denoted by w) is shown below inEq. (1).

ρA∂2w∂t2 + c

∂w∂t

+EI∂4w∂x4 + feV 2 = 0 (1)

In Eq. (1), fe denotes the electrostatic force on the beamfrom the electrode. The analytical expression of this forcewas determined in [6] by using a fitting expression from a



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finite element simulation. We verified the fitting expressionby making a COMSOL simulations of the 2D section of theelectrode.

fe(w) =asinh(σ w

h )

coshp(σ wh )

H(x− xe) (2)

In Eq. (2), a, σ, and p are fitting parameters that dependon the geometry of the beam, and H is the Heaviside stepfunction, with the values of the fitting parameters given inTable 1.

Equation (1) is first non-dimensionalized. These substi-tutions are as follows:

x = x/L

g = g/h

w = w/h

t = t√

EI/ρAL4

c∗ = c√

L4/EIρA

fe = fe/a

β = aL4V02/EIh

β is a new non-dimensional parameter that represents theconstant coefficients of the electrostatic forcing function.The non-dimensional equation of motion is shown in Eq. (3).

∂2w∂t2 + c∗

∂w∂t

+∂4w∂x4 +βV 2 fe = 0 (3)

In both the non-dimensional substitutions and Eq. (3),the non-accented variables denote non-dimensional quanti-ties. Equation (3) is then solved using the Galerkin decom-position procedure. Using this method, the deflection of thebeam is approximated as

w(x, t)≈n

∑i=1

qi(t)φi(x) (4)

where φi(x) are the mode shapes of the beam, and qi(t)are the time dependent generalized coordinates, which mustbe solved numerically. The mode shapes can be obtainedby solving the eigenvalue problem associated with Eq. (3)as will be described in the next section. Once the modeshapes are obtained, one can substitute Eq. (4) into the non-dimensionalized equation of motion (Eq. (3)), multiply by φiand integrate over x between 0 and 1 to get

miqi + c∗miqi + kiqi +βV 2 fi = 0 (5)

This is done to reduce the partial differential equation(PDE) into a set of decoupled ordinary differential equations(ODE) using the orthogonality of the mode shapes. In Eq.(5), mi, ki, and fi are defined in Eq. (10).

2.2 Flexible SupportThe addition of a flexible support is taken into account

by the mode shapes of the beam (φ(x)). Specifically, thisintroduces a new set of boundary conditions for φ(x) givenbelow [12]. Figure 2 shows the comparison between the flex-ible and fixed support.

φ′′(0)− Lkr

EIφ′(0) = 0

φ′′′(0)− L3kt

EIφ′(0) = 0

φ′′(L) = 0

φ′′′(L) = 0

(6)

In Eq. (6) kr and kt are the rotational and translationalstiffness of the flexible boundary respectively. Two non-dimensional variables, Rr and Rt , are defined for simplicityand are shown in Eq. (7).

Rr =Lkr

EIRt =

L3kt

EI(7)

Once the boundary conditions have been defined, themode shapes can be computed. This is done by setting boththe voltage and damping of Eq. (3) equal to zero, and usingthe Galerkin procedure (separation of variables) to reduce thePDE into two ODEs. A generalized mode shape

φ = Acos(αx)+Bsin(αx)

+C cosh(αx)+Dsinh(αx) (8)

is used, where A, B, C, and D are constants, and α is thesquare root of the non-dimensional natural frequency. Sub-stituting this back into the boundary conditions of Eq. (6)yields the system of equations shown in Eq. (9).

−α −Rr α −RrRt −α3 Rt α3

−cos(αL) −sin(αL) cosh(αL) sinh(αL)sin(αL) −cos(αL) sinh(αL) cosh(αL)

ABCD

= 0

(9)For a non-trivial solution, the determinant of the ma-

trix on the left side must be zero, which yields a characteris-tic equation for α, the eigenvalue. Solving numerically, theeigenvalues are obtained and used in Eq. (9) to find A, B, C,and D. Mode shape, φ, is then normalized to have a maxi-mum of one. Using these mode shapes and the orthogonalitycondition, one can obtain the coefficients of mi, ki and fi inthe decoupled equation of motion (Eq. (5)):



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mi = δi j

∫ 1

0φ jφidx

ki = δi j

∫ 1

0φ jφ

IVi dx

fi =∫ 1

xe

feφ jdx

(10)

where δi j is the Kronecker delta, that is zero when i 6= jand is one when i = j. Equation (5) is then solved numeri-cally in MATLAB using the 4th order Runge-Kutta methodwith the ode45 solver [24].

2.3 Numerical ResultsThe effect of the flexible support on the parametric res-

onance of the beam was studied for a one mode model. Thedamping was estimated as

c =λ2

1Q

(11)

where c = c∗m1 in Eq. (5) for one mode model. In Eq.(11), λ2

1 is the first natural frequency of the cantilever andthe flexible support, and Q is the quality factor. The qualityfactor experimentally determined in [6] for the fixed supportwas used in Eq. (11) as an estimate. This value was cho-sen due to the similarity between the two resonators whenthe boundary stiffness is sufficiently large. It should also benoted that due to the single layer design of the beam andelectrode, squeeze film damping is almost eliminated, and amajor part of the total damping is due to viscous and supportdamping. It is assumed that the change in support dampingas boundary stiffness decreases can be roughly accounted forby Eq. (11). This can be visualized in the damping profile ofFigure 3.

The beam was also given a very small initial offset to ini-tiate parametric resonance. This is necessary because whenthe beam is not deflected, the net electrostatic force is zero.

To verify the accuracy of the presented model, first Rrand Rt were set to 10,000. At Rr = Rt = 10,000 and above,the rotation and translation of the flexible boundary are neg-ligible and the beam behaves like a simple cantliever. Thiswas determined by plotting m, c, and k values for increas-ing boundary stiffness and looking for convergence to the m,c, and k values for the fixed boundary. Figure 3 shows thisconvergence to those of the fixed support reported in [6].

To study the effect of a rotational and translationalspring at the support, three cases are investigated. First Rr isdecreased while Rt is held fixed (Case I). This is the equiva-lent of a pure rotational spring with no translational move-ment of the supported end. Next, Rt is varied and Rr isheld fixed (Case II), which is similar to an ideal translationalspring. Lastly, Rr and Rt are equated and analyzed at dif-ferent values to simulate the complete flexible support (Case

III). The frequency response for Case I, II, and III, at 100volts AC can be seen in Figure 4, 5, and 6 respectively.

All three cases have similar effects on the frequency re-sponse domain with Case III being the most pronounced.First, there is a significant downward shift in the frequencyresponse due to the reduction of the overall stiffness of thesystem. Second, the maximum amplitude increases when theboundary stiffness decreases up until Rr and/or Rt reach athreshold. An increase of 66 percent in the maximum am-plitude is observed when the stiffness reaches the thresh-old value (50µm at Rr = Rt = 10 compared to 30µm atRr = Rt = 10,000 for the fixed support). A time responseat its maximum amplitude in Figure 6 is shown in Figure7. Below the threshold value, the response becomes eitherodd period or chaotic (depending on the frequency) as willbe discussed shortly. This can be seen in Figure 4, 5, and 6when the frequency response curve, obtained from the max-imum amplitude, is no longer smooth and appears jagged. IfRr and Rt are decreased further, the response becomes com-pletely chaotic and does not reach steady state.

The frequency range where parametric resonance isspread is also increased as Rr and/or Rt are decreased. AtRr = Rt = 10 the range is about 7000 Hz compared to about5000 Hz for the fixed boundary. Higher order parametric res-onances were also observed as the stiffness varies (not shownin Figure 4, 5, and 6). Figure 8 shows the higher order para-metric resonances for the case of Rr = Rt = 10. As seen inFigure 8, higher order parametric resonances occur at 6.75kHz, 4.5 kHz, and 3.375 Hz respectively. We observed thathigher order parametric resonances move closer to the pri-mary resonance as the boundary stiffness is decreased.

There are slight differences between the three cases.Case I does not increase the amplitude as much as Case IIor III. Case II has some increase in maximum amplitude butnot as large as Case III. Even though all three cases expe-rience a chaotic response when the boundary stiffness goesbelow a threshold, the chaotic attractor is the largest in CaseII, meaning the translational stiffness has a larger effect onthe size of the chaotic attractor compared to the rotationalstiffness.

As mentioned above, in all three cases, the response ofthe resonator becomes chaotic when the boundary stiffnessgoes below a threshold. Around the threshold stiffness, asmall variation in peak amplitude of the steady state responseappears. Decreasing the boundary stiffness below the thresh-old causes the variation to become more apparent and chaoticresponse occurs.

To verify the response has a chaotic nature, we plottedthe bifurcation diagram for the last 50 cycles. The bifur-cation diagrams shown in Figures 9 and 10 are plotted forRt = 10,000, Rr = 2 and Rr = 25 respectively. The diagramis obtained from recording the positive displacement as thephase portraits cross the zero velocity axis. As can be seenin Figure 9, in certain regions of the bifurcation diagram, thechaotic attractor size is large. While other areas of the curveseem periodic, the chaotic attractor is just smaller in size. Infact, for most of the resonant frequency curve, in the case ofRr = 2 (Figure 9), the motion is chaotic. As the boundary



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stiffness is increased, the chaotic behavior disappears as canbe seen in Figure 10.

To prove the motion is indeed chaotic, the informationentropy and dimension were calculated. Information entropycalculates the unpredicitbility of a system. For a non-chaotic,deterministic system, the information entropy is zero becausethe system is fully predictable. The information dimension,calculated from the information entropy, is a quantitativemeasure of the fractal behavior of the attractor. Non-integervalues of the information dimension show the attractor isstrange, while a non-zero information entropy shows a lossof predictability of the system, indicating chaos [25].

To calculate the information entropy and dimension, aPoincare section was taken by sampling points at the excita-tion frequency starting at the maximum displacement of thelast cycle (Figure 11). The Poincare section of the attrac-tor was then divided into small squares of non-dimensionalsize 0.002 × 0.002. The number of points that fall in eachsquare was counted and divided by the total number of sam-ple points to calculate the probability of a point landing inthat square for the set of data. This probability distribu-tion was used with Eq. (12) to calculate the information en-tropy [25].

I(ε) =−N

∑i=1

Pi logPi (12)

In Eq. (12), N is the total number of squares in the phaseplane, ε is the area of one square, and Pi is the probability forthe ith square. The information entropy was then used in Eq.(13) to find the information dimension.

di =−I(ε)logε

(13)

Figure 12 shows the information dimension for variousinitial conditions for the case of Rt = 10000, Rr = 2. Theinformation dimension is approximately 0.586. As this isa non-integer value, the attractor is shown to exhibit frac-tal behavior. An information dimension of 0.586 also corre-sponds to an information entropy of approximately 10.5 bits.This proves the system is chaotic. The information dimen-sion should be consistent for a chaotic attractor for any initialcondition. Figure 12 shows that the information dimension isconsistent for various initial conditions that result in motionon the chaotic attractor.

The chaotic behavior arises from the nonlinear mod-ulation of the effective stiffness. The high compliance ofthe beam due to the decreasing boundary stiffness increasesthe influence of this nonlinearity, which ultimately results inchaos. This can be verified through experiments by measur-ing the information entropy of the tip deflection.

As mentioned above, the vast majority of the resonantfrequency curve is chaotic. However, at certain frequencies

the motion is periodic. This shows the system is experiencingintermittent chaos, snapping between period 1-7 motion andchaos. These n-period regions can be seen in the bifurcationdiagrams and Poincare maps. Figure 13 shows a zoomed inportion of Figure 9, showing period 3 and 5 behaviors. De-spite intermittent chaos, no period doubling was observed atthe onset of chaotic motion. A time response in the period 5region, shown in Figure 14, illustrates the expected 5 distinctpeaks.

The transition from the periodic to the chaotic region ofFigure 13 also shows other interesting behavior. The evolu-tion of Poincare section from period 5 to chaos is also evidentin Figure 11, which shows 5 prominent lumps of points thatare spreading in the phase plane. As the system moves fromperiod 5 to chaos, the variation in peak amplitude increaseslocally around the peak points from the period 5 regime. Asthe frequency is increased, these points spread out and thePoincare section becomes more uniform and spread.

After the dynamic response was analyzed at a high volt-age (100V), it was then analyzed again at low voltages to de-termine its threshold for PR. Specifically, the applied voltagewas incrementally increased by 1V, starting below the thresh-old voltage, to see the minimum voltage required for the ap-pearance of parametric resonance. As an example, choosingthe stiffness values that resulted in the largest amplitudes inFigure 6 (Rr = Rt = 10), and reducing the AC voltage yieldsthe results shown in Figure 15.

We recorded the lowest voltage required for parametricresonance to appear in all three cases. Case II has the lowestthreshold voltage at about 10V. Case I and III have a thresh-old voltage of 12V and 13V respectively. The threshold volt-age is reduced from 27V for the fixed support case [6]. Therespective threshold voltages for all cases, including the fixedsupport, have the same amplitude as well (around 5 µm). Allthree cases significantly outperform the fixed support, eventhough the differences between each was slight. The flexiblesupport also has a significant increase in amplitude at lowvoltages, e.g. at 16V for Case III, the flexible support hasapproximately the same amplitude as the fixed boundary at40V. However, the frequency at which parametric resonanceshappens for the flexible support is one third of that of thefixed support, which decreases the rise time of the response.

Overall, Case III has the most desirable effect on thefrequency response achieving the highest amplitudes whilestill cutting the threshold voltage by over half as comparedto the fixed support.

3 Flexible Support Design and ResultsAfter the ideal model was analyzed, a physical design

that could attain the results of the ideal case (Rr = Rt = 10)was determined. As mentioned before, a crab-leg spring wasused and can be seen in Figure 1. The rotational and trans-lational stiffness were evaluated using Castigliano’s secondtheorem [26], and were a function of the spring geometry.



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kr =

(S

GJ+

L1

EI

)−1

(14)

kt =

(L3

1 +S3

3EI+

L1S3

GJ

)−1

(15)

In Eqs. (14) and (15), I and J are the moment of inertiaand polar moment of inertia for the cross section of the springrespectively. The stiffness values of Eq. (14) and (15) werethen substituted into Eq. (7) and non-dimensional Rr was setequal to the non-dimensional Rt value. This new equationwas then used to find geometry that resulted in equal Rr andRt values.

To satisfy equality of Rr and Rt , the geometry needs tobe very large however. To obtain a reasonable resonator size,Rr and Rt values were un-equated and given some set dif-ference. The larger the difference between Rr and Rt , thesmaller the geometry, and vice versa. Therefore, a middleground between stiffness and geometry was found to maxi-mize the effect of the flexible boundary, with minimal geom-etry.

Geometric parameters of the support, along with the re-sulting Rr and Rt values are shown in Table 2 for severaldesigns, with parameters shown in Figure 1. The material ofthe spring is assumed to be the same material as that of thebeam.

As can be seen in Table 2, smaller length values of S re-sult in a high translational stiffness without having much ef-fect on the rotational stiffness, which can be confirmed fromEq. (14) and (15). This is mainly what causes the large dif-ference between Rr and Rt values. A small S is desirable be-cause it determines the width, and therefore the overall sizeof the resonator. As S is decreased, the model becomes sim-ilar to the rotational model (Case I) as described above. Fig-ure 16 shows the frequency response for design 3, and Figure17 shows the frequency response at low voltages, includingthe threshold voltage.

Compared to the fixed support, the flexible support fromdesign 3 offers a 50 percent increase in the maximum am-plitude (45µm compared to 30 µm). We also observe theappearance of higher order parametric resonances with pri-mary resonance around 15.5 kHz. The threshold voltage wasalso decreased by 52 percent (15V compared to 27V).

Among all designs, design 4 that has a much smaller ro-tational stiffness compared to translational stiffness (similarto case I in Figure 9), yields a chaotic response. By increas-ing L1 to 100µm, the rotational boundary stiffness drops be-low the threshold value, and chaos occurs. If a serpentinespring is used instead of crab-leg used here, it could resultin a lower rotational stiffness and thus would be even moresusceptible to a chaotic response. Although chaos couldoccur for case II (pure translational spring), the geometryof the resonator to create a pure translational spring at the

boundary, with a stiffness below the threshold, would needto be very large and thus is unlikely to occur in a real-worldMEMS device, and thus not considered.

4 ConclusionThis paper demonstrates the effect of a flexible support

on a parametrically excited electrostatic MEMS resonator.The resonator uses weaker electrostatic fringe fields to actu-ate a cantilever micro-beam without the risk of pull-in, al-lowing for high amplitudes of vibration. The flexible sup-port significantly decreases the threshold voltage, creating amore energy efficient resonator. The effect of boundary stiff-ness on the magnitude and width of the frequency responseis thoroughly investigated. We observe that decreasing theboundary stiffness increases the output amplitude leadingto a high signal to noise ratio. If the boundary stiffness isdecreased below a threshold, it can yield chaotic behavior,which has been verified by the information entropy and di-mension. The chaotic response may not be desired if theresonator is used as a sensor as it could complicate the sen-sor circuitry and hence should be avoided. A compromiseneeds to be made for the stiffness value to achieve the maxi-mum amplitude and to avoid the chaotic region. An optimaldesign for the flexible support is described, which yields a50 percent increase in the maximum amplitude and a 50 per-cent decrease in the threshold voltage for the appearance ofparametric resonance.

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[15] Turner, K. L., Miller, S. A., Hartwell, P. G., MacDon-ald, N. C., Strogatz, S. H., and Adams, S. G., 1998.“Five parametric resonances in a microelectromechan-ical system”. Nature, 396(6707), pp. 149–152.

[16] Kobrinsky, M. J., Deutsch, E. R., and Senturia, S. D.,2000. “Effect of support compliance and resid-ual stress on the shape of doubly supported surface-micromachined beams”. Journal of Microelectrome-chanical Systems, 9(3), pp. 361–369.

[17] Pakdemirli, M., and Boyaci, H., 2002. “Effect of Non-Ideal Boundary Conditions on the Vibrations of Con-tinuous Systems”. Journal of Sound and Vibration,249(4), pp. 815–823.

[18] Rinaldi, G., Packirisamy, M., and Stiharu, I.,2008. “Boundary characterization of MEMS structuresthrough electro-mechanical testing”. Sensors and Ac-tuators, A: Physical, 143(2), pp. 415–422.

[19] Alkharabsheh, S. a., and Younis, M. I., 2013. “Dynam-ics of MEMS arches of flexible supports”. Journal ofMicroelectromechanical Systems, 22(1), pp. 216–224.

[20] Alkharabsheh, S. A., and Younis, M. I., 2011. “TheDynamics of MEMS Arches of Non-Ideal BoundaryConditions”. Vol. 7, Proceedings of the ASME DesignEngineering Technical Conference, pp. 197–207.

[21] Boyaci, H., 2006. “Vibrations of stretched dampedbeams under non-ideal boundary conditions”. Sad-hana, 31(1), pp. 1–8.

[22] Rinaldi, G., Packirisamy, M., and Stiharu, I., 2007.“Quantitative Boundary Support Characterization forCantilever MEMS”. Sensors, 7(10), pp. 2062–2079.

[23] Zhong, Z. Y., Zhang, W. M., and Meng, G., 2013.“Dynamic characteristics of micro-beams consideringthe effect of flexible supports”. Sensors (Switzerland),13(12), pp. 15880–15897.

[24] MathWorks, 2016. “MATLAB Function Reference”.pp. 6613–6614.

[25] Moon, F. C., 1992. Chaotic and Fractal Dynamics. Wi-ley Interscience.

[26] Budynas, R., and Nisbett, K., 2011. Mechanical Engi-neering Design, 9 ed. Mcgraw-Hill, New York.



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List of Figures1 Layout of the resonator with cross section of the beam and top down view of electrode and beam tip. . . . . 92 Fixed vs. flexible support. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 The convergence of non-dimensional mass (m), damping (c), and stiffness (k) values to the fixed boundary

case (dashed lines) as Rr and Rt increase. Good convergence occurs at approximately 104 and above. . . . . 114 Case I, Rr frequency response (Rt = 10,000) at 100V AC. Numbers next to lines indicate the value of Rr.

Solid lines indicate downsweep and dashed lines indicate upsweep. Frequency step of 20Hz. . . . . . . . . 125 Case II, Rt frequency response (Rr = 10,000) at 100V AC. Numbers next to lines indicate the value of Rt .

Solid lines indicate downsweep and dashed lines indicate upsweep. Frequency step of 20Hz. . . . . . . . . 136 Case III, Rr = Rt frequency response at 100V AC. Numbers next to lines indicate the value of Rr and Rt .

Solid lines indicate downsweep and dashed lines indicate upsweep. Frequency step of 20Hz. . . . . . . . . 147 Rr = Rt = 10 time response in steady state regime with an approximately 50 µm amplitude. Driven at 10880

Hz at a voltage of 100V. An initial displacement of 40 µm was used to reach the higher energy state in thebacksweep region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

8 Frequency response of Rr = Rt = 10 showing higher order parametric resonances. Frequency step of 20Hz. 169 Bifurcation diagram for Rt = 10000, Rr = 2 showing peak amplitude against frequency for the last 50 cycles

in the steady state regime. Backsweep only. Includes both primary and fundamental parametric resonances.Frequency step of 10 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

10 Bifurcation diagram for Rt = 10000, Rr = 25 showing peak amplitude against frequency for the last 50 cyclesin the steady state regime. Backsweep only. Frequency step of 10 Hz. . . . . . . . . . . . . . . . . . . . . 18

11 Poincare section of the chaotic attractor for Rt = 10000, Rr = 2, and f = 12 kHz. A total of 1500 points weresampled. The simulation was run for 2000 cycles, with sample points starting after cycle 500 to ensure thesystem was on the chaotic attractor, and had negligible transient effects. . . . . . . . . . . . . . . . . . . . 19

12 Information Dimension for Rr = 2, Rt = 10000 at f = 12 kHz. Initial conditions are non-dimensional. . . . 2013 Zoomed in portion of the Rr = 2 bifurcation diagram showing intermittent chaos and period 5 behavior. 0.1

Hz frequency step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2114 Time response of Rr = 2 in period 5 regime at 12.44 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2215 Backsweep frequency response of Rr = Rt = 10 for low voltages. Numbers next to lines indicate voltage

level. Frequency step of 5Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2316 Frequency response of support design 3 as listed in Table 2: Rr = 5.33 and Rt = 2203.26, at 100V AC. Solid

lines indicate downsweep and dashed lines indicate upsweep. Frequency step size was 20Hz. . . . . . . . . 2417 Backsweep frequency response of design 3, with Rr = 5.33 and Rt = 2203.26. Numbers next to lines indicate

voltage level. Frequency step of 5Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25



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Fig. 1. Layout of the resonator with cross section of the beam and top down view of electrode and beam tip.



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Fig. 2. Fixed vs. flexible support.



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100 101 102 103 104 1050

0.25

0.5

0.75

Mass

100 101 102 103 104 1050

0.01

0.02

0.03

Dam

pin

g

100 101 102 103 104 105

Rr & R

t values

0

1.5

3

Stiffness

Fig. 3. The convergence of non-dimensional mass (m), damping (c), and stiffness (k) values to the fixed boundary case (dashed lines) asRr and Rt increase. Good convergence occurs at approximately 104 and above.



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5 10 15 20 25 30 35

Frequency (kHz)

0

5

10

15

20

25

30

35

40

Am

plit

ude (µ

m)

2 610 15 25

50 10,000

Fig. 4. Case I, Rr frequency response (Rt = 10,000) at 100V AC. Numbers next to lines indicate the value of Rr . Solid lines indicatedownsweep and dashed lines indicate upsweep. Frequency step of 20Hz.



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5 10 15 20 25 30 35

Frequency (kHz)

0

10

20

30

40

50

Am

plit

ude (µ

m)

26

10

15 2550 10,000

Fig. 5. Case II, Rt frequency response (Rr = 10,000) at 100V AC. Numbers next to lines indicate the value of Rt . Solid lines indicatedownsweep and dashed lines indicate upsweep. Frequency step of 20Hz.



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5 10 15 20 25 30 35

Frequency(kHz)

0

10

20

30

40

50

Am

plit

ude (µ

m)

10,000

6

10

15

2550

Fig. 6. Case III, Rr = Rt frequency response at 100V AC. Numbers next to lines indicate the value of Rr and Rt . Solid lines indicatedownsweep and dashed lines indicate upsweep. Frequency step of 20Hz.



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63.4 63.6 63.8 64 64.2

Time (ms)

-50

-25

0

25

50

Dis

pla

cem

ent (µ

m)

Fig. 7. Rr = Rt = 10 time response in steady state regime with an approximately 50 µm amplitude. Driven at 10880 Hz at a voltage of100V. An initial displacement of 40 µm was used to reach the higher energy state in the backsweep region.



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4 6 8 10 12 14 16 18

Frequency(kHz)

0

10

20

30

40

50

Am

plit

ude (µ

m)

Fig. 8. Frequency response of Rr = Rt = 10 showing higher order parametric resonances. Frequency step of 20Hz.



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6 8 10 12 14 16 18

Frequency (kHz)

0

5

10

15

20

25

30

35

40

45

Peak A

mplit

ude (µ

m)

Fig. 9. Bifurcation diagram for Rt = 10000, Rr = 2 showing peak amplitude against frequency for the last 50 cycles in the steady stateregime. Backsweep only. Includes both primary and fundamental parametric resonances. Frequency step of 10 Hz.



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23 24 25 26 27 28 29 30

Frequency (kHz)

0

5

10

15

20

25

30

35

Peak A

mplit

ude (µ

m)

Fig. 10. Bifurcation diagram for Rt = 10000, Rr = 25 showing peak amplitude against frequency for the last 50 cycles in the steady stateregime. Backsweep only. Frequency step of 10 Hz.



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8 9 10 11 12 13

Displacement (µm)

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Velo

city (µ

m/s

)

Fig. 11. Poincare section of the chaotic attractor for Rt = 10000, Rr = 2, and f = 12 kHz. A total of 1500 points were sampled. Thesimulation was run for 2000 cycles, with sample points starting after cycle 500 to ensure the system was on the chaotic attractor, and hadnegligible transient effects.



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2 2.05 2.1 2.15 2.2 2.25

Initial Condition

0.54

0.56

0.58

0.6

0.62

Info

rmation D

imensio

n

Fig. 12. Information Dimension for Rr = 2, Rt = 10000 at f = 12 kHz. Initial conditions are non-dimensional.



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12.3 12.35 12.4 12.45 12.5

Frequency (kHz)

7

8

9

10

11

12

13

Peak A

mplit

ude (µ

m)

Fig. 13. Zoomed in portion of the Rr = 2 bifurcation diagram showing intermittent chaos and period 5 behavior. 0.1 Hz frequency step.



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79 79.2 79.4 79.6 79.8 80 80.2

Time (ms)

-10

-5

0

5

10

Dis

pla

ce

me

nt

(µm

)

Fig. 14. Time response of Rr = 2 in period 5 regime at 12.44 kHz.



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10.85 10.9 10.95 11 11.05 11.1 11.15 11.2

Frequency(kHz)

0

2

4

6

8

10

12

Am

plit

ude (µ

m)

14 V

13 V

15 V16 V

19 V

18 V17 V

Fig. 15. Backsweep frequency response of Rr = Rt = 10 for low voltages. Numbers next to lines indicate voltage level. Frequency step of5Hz.



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6 8 10 12 14

Frequency (kHz)

0

5

10

15

20

25

30

35

40

45

Am

plit

ude (µ

m)

Fig. 16. Frequency response of support design 3 as listed in Table 2: Rr = 5.33 and Rt = 2203.26, at 100V AC. Solid lines indicatedownsweep and dashed lines indicate upsweep. Frequency step size was 20Hz.



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15.6 15.65 15.7 15.75 15.8 15.85 15.9

Frequency (kHz)

0

2

4

6

8

10

Am

plit

ude (µ

m)

15 V16 V 17 V 18 V

19 V

Fig. 17. Backsweep frequency response of design 3, with Rr = 5.33 and Rt = 2203.26. Numbers next to lines indicate voltage level.Frequency step of 5Hz.



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List of Tables1 Beam parameters of MEMS resonator given in [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 Geometry and stiffness of flexible support. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28



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Table 1. Beam parameters of MEMS resonator given in [6].

Parameter Symbol Value

Beam Length (µm) L 500

Beam Width (µm) b 16

Beam Height (µm) h 5

Electrode Gap (µm) g 5

Electrode Length (µm) Le 150

Elastic Modulus (Gpa) E 169

Density (kg/m3) ρ 2330

Poisson’s Ratio ν 0.3

Quality Factor Q 150

Fitting Parameter a 1.3∗10−6

Fitting Parameter σ 1

Fitting Parameter p 1.45



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Table 2. Geometry and stiffness of flexible support.

Parameter Design 1 Design 2 Design 3 Design 4

S (µm) 200 100 30 30

L1 (µm) 20 30 30 100

t (µm) 4 3 4 5

w (µm) 10 20 10 10

Rr 7.999 4.499 5.33 3.12

Rt 14.86 97.747 2203.26 206.56



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Parametrically Excited Electrostatic MEMS Cantilever Beam ... · Parametrically Excited...

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Transcript of Parametrically Excited Electrostatic MEMS Cantilever Beam ... · Parametrically Excited...