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8/12/2019 Multiple Mode Resonator

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MULTIPLE ODEMICROMECHANICALESONATORSReid A. Brennen, Albert P. Pismo, and William C. Tang

Berkeley Sensor 6 Actuator CenterElectronics Research LaboratoryDepartment of Electrical Engineering and Computer ScienceUniversity of California Berkeley CA 94720

ABSTRACTIn this paper are described two resonant micromechanicalstructures that have been designed, fabricated, and tested and whichexhibited multiple modes of v ibration. The first had a cantileveredinertial mass that was actuated by a curved-comb electrostatic-driveattached near the root of the cantilever. The second structure had astraightamb electrostatic-drive with a folded flexure suspension thatwas used to actuate the cantilevered inertial mass. The first structureexhibited 2 distinct vibration modes, and the second structure actuallyexhibited 3 distinct vibration modes. An analytic dynamic model hasbeen developed and it predicted the vibrational mode shapes of thestructures. Tests of the fabricated structures have demonstrated thatpeak-to-peak lateral displacements greater than 10 microns werefeasible. Further, peak-to-peak angular displacements greater than100 have been measured dur ing second mode vibration. The resonantfrequencies of the structures varied from 1.7 to 33 kHz depending onstructure geometry. Cantilever structures with overhang lengths asgreat as 864 microns have been fabricated and operated with noperceptible contact between the inertial mass and the substrate.INTRODUCTIONMicro electromechanical resonant systems hold out the promiseof providing extremely fine position control at very high mechanicalfrequencies with extremely low power input. Some microresonatorshave been shown capable of achieving high frequency resonance (750kHz) at high Q (above 35,000), when fabricated of polycrystallinesilicon and operated in a vacuum [l]. Other microresonators have beendesigned for relatively large linear motion of a rigid body vibrationmode asopposed to a continuum vibration mode) and these, too, havebeen shown to achieve significant values of Q 49,000) for largeamplitude motion 20 micrometers) at 31 kHz as reported by [21, I31 as

well asby [41.The common feature of all these microresonators is that theirvibration mode is purely linear. This is an artifact of the mechanicaldesign featuresof the flexural suspension systems and the mechanics ofthe boundary conditions where the flexures are anchored to thesubstrate. Clearly the designers have explicitly chosen mechanicaldesigns that deliberately preclude rotational motion. Other designshave been tailored to flexural suspensions suitable for purely angularresonance,and such a device is described in [21.In this esearch effort,a microresonator has been designed witha flexural suspension that admits resonances in all of translational,rotational, and coupled translational/rotationalmodes of vibration.These modes are not only well-separated in frequency, but can beselectively excited and analytically predicted. This makes it possibleto design a new class of resonant structure micromotors with selectablemodes of operation.Micromotors that attempt to achieve rotational motionwithout the use of flexures must implement some sort of mechanicalbearing in which surface-to-surface contact friction is inevitable [5], [61.Such micxomotors have the advantage of limitless rotation, but this isobtained at the cost of friction and wear. Rotational resonant-structuremicromotors may not provide full rotation motion, but operate at highfrequency with substantial Q in the complete absence of mechanicalfriction and wear. Until research in the complete and controlledlevitation of armatures in micromechanical rotation motors provides aworkable engineering design [71, rotational resonant-structuremicromotors s e m o offer the only solution to high-frequency, zero wearrotational motion.CH2832-4/90/0000-0009 01.0001990 IEEE 9

In this research, two mechanical designs for multiplemode,micromechanical resonators have been developed, fabricated,operated, and quantified experimentally. Analytic, dynamic modeshave been developed for each, and it is Seen that these models predictvibration mode shapes and vibration frequencies.

DESIGNThe two basic designs of the multiple mode resonating structuresare shown inFigs.1and 2. Both designs are modular two componentsystems that consist of a semaphore and a driving base . Thesemaphore section in both designs is the resonator proper, and is madeu p of a single cantilever beam with an end mass. The beam can beattached at one edge of the plate mass, or, as shown inFigs.1and 2, thebeam can be attached at the gravity center of the mass. The other endof the semaphore beam is attached to the drive-base, of which twodistinct designs are presented below. The drive-base in each caseconsists of an electrostatic comb drive [31 which is supported by afl ex w suspension. The electrostatic driveused two comb elements, onewhich was used for driving the structure and one which was used to

A canhkver design, as opposed to some variation of a resonantbridge form [3], was used for the semaphore for several reasons. Acantilever be mhas 64 imes the compliance of a n equal length beamthat is fixed at the ends, and thus, he cantilever design affords greatercompliance in a shor ter distance. Further, the canti lever design is acompletely released structure that does not retain any residual stressesinduced by the fabrication process. Lastly, this design allows anadditional degree of freedom in the semaphore mass, a rotationalvibration, which can be exploited.

The first of the two designs for the drive-base (Fig. 1 is alinear drive design that uses a folded beam suspension [3] support.

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Fig. 1. Photograph of the h e a r drive multimode resonator structure.Note Gshaped breakaway upport attached to the mas8 to the right.

Fig. 2. Photograph of the angulardrive multimode resonator structure.


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Fig. 3. The center of curvature of the curved comb teeth is set at U3 thedistancefrom thebeam root to the comb mass. Fig. 4. SEM micrographof the electrostatic comb drive portion of thelinea drive structure.

This design forces the drivebase to undergo translational motion only,and the semaphore is therefore driven at its beam root only by atranslational motion. In contrast, the design shown in Fig. 2is anangular drive design that employs a drive-base with coupledtranslational and rotational motion. One way of regarding this seconddesign is to consider it a staged pair of cantilever beams, one each forthe drive-base and the semaphore. The complexity of the motion ofthe drive-base forces the use of curved comb teeth with varying radiusof curvature, since displacements are large enough that kinematicnonlinearities must be accounted for. Empirical calculations were usedto determine that to provide the maximum clearance between thedrive-base comb teeth and the fixed teeth as the drive moves, thecenter of curvature of the comb eeth must be set to 2/3 thedistance fromthe beam/comb-structure unction to the supportingbeam root, as shownin Fig. 3. Attaching the comb teeth directly to the beam, without localstiffening of the beam at the root of the comb would allow a largerdeflection but the resulting design would not allow the use of theempirical tooth curvature result stated above at high resonantfrequencies. This design would be undesirable since there would becontactbetween the fixed- and moving-comb teeth, effectively shortingout the electrostatic comb drive.The two component system allows a mechanical impedancematchmg in that the high-amplitude semaphore motion can be driveneither by large or small base motions. By tuning the physicaldimensions and masses of the device components and selecting whichresonant mode to excite, the maximum response of the semaphore canbeachieved with either large or small base amplitude motion. Since thesemaphore cantilever beam is attached to the gravity center, theangular motion of the mass has been decoupled from the translationalmotion of the beam, and likewise, the translational motion of the massfrom the angular motion of the beam. This decoupling makes itpossible to selectively excite either translational or angular vibrationmodesof themass.In all structures fabricated, the beam widths as well as thebeam thicknesses were nominally 2p. he semaphore mass was 116by 196p.m. Each tooth of the electrostatic comb drive was 2 by 2 pm incross section and was separated laterally from the fixedcomb teeth by2 pm, as shown inFigs.4and 5.Due to the arge sizeof some of the resonant structures, the totalspan of the cantilevered section reaches as much as 864 pm and atemporary restraint was necessary to prevent structure deflection anddamage during the fabrication process. All structures are suspendedabove the substrate at a 2 pm height. Temporary breakaway supports,which c n be seen inFigures1and 2, were attached to the outside of themass of the semaphore. These breakaway supports were used toprovide strength to the structure during fabrication so that the longstructures were not destroyed during wet etching and rinsing. Afterfabrication, the breakaway support was mechanically removed byphysically rupturing it with two probes.

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Fig. 5. SEM miaogmph of the electrostatic comb drive portion of theangulardrive structurr .

DYNAMIC MODELLINGTwo separate dynamic models were developed to simulate thebehavior of the two types of resonant, multimode structures, since eachof the drive-bases were fundamentally different for the two designs.The linear-motion drive-base structure with the folded flexuresuspension (Fig. 6) had 4 degrees of freedom 3 translational and 1rotational. The base drive of thisdesign had 2 degreesof freedom,onebeing the comb structure mass, the other being the fo ld piece in thefolded flexure. As noted above, the semaphore section itself had 2degrees of freedom, one translational and one rotational, both of whichwere measured from the equilibrium position of the center of massof theplate mass. The second structure with the angular drive-base (Fig.7)also had 4 degrees of freedom, but these were d ivided into 2translational and 2 rotational categories. The semaphore for thisdesign was identical to the semaphore used in the linear drivebasedesign described above, but the driver had 1 translational and 1rotational degree of freedom.For the purpose of analyzing the two designs and predictingmode shapes and vibration frequencies, it was assumed that thedisplacements were small compared to the beam lengths. Thisassumption would cause some inaccuracies since the experiment didexhibit large amplitude motion, but this inaccuracy tumed out to besmall. The mass of the beams was ignored in the dynamic model aswere any geometric non-linearities such as non-vertical beamsidewalls. In models of both resonators, the flexures were assumed tobend in quasi-static modes; he dynamic deflection shapes of the beamwere assumed to closely correspond to the static ones. Although thisassumption would be expected to break down at very high fquencies,

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Fig. 6. Schermtic drawing of a linear drive multimode resonatings t r u m . The p o w ~efine the degreesof freedom for he structure.

Fig. 7. Schematic drawing of an angular drive multimode resonatings t ruchur . Theauow define the degrees of freedom forthe structure.

the experimental results indicated that the primary vibrationfrequencies were not highenough for thisassumption o fail.In these dynamic models, only viscous damping under themasses was considered, even though this was known to result in anunderestimateof the actual amount of air damping present a t ambientatmospheric pressures [31. The flow of the air under the semaphoremass was also assumed to be fully developed (i.e. possess a linearvelocity prome), and the damping due to drag on the top of the plate(facing away from the substrate) was assumed to be small in thissimulation, aswas the drag due to the air flow past the edges of theplates and beams. The internal damping in these structures was verysmal l compared to that due to the air and for the purposes of predictingmode shapes nd vibration fque nci es was likewiseassumed to be zero.Since the structure had a constant thickness, and damping wasproportional to the area under the mass, the damping acting on thetranslational mode of vibration was proportional to the mass of eachcomponentand thedamping of the rotational mode was proportional tothe polar moment of inertia of each component. This proportionalitybetween mass and damping comes from the basic assumption ofdeveloped flowand theabsence of centrifugal motion of the air betweenthe resonatingmassand substrate.1 1 1

Taking a Newtonian approach to dynamics, the inertial forcesand momenta were set equal and opposite to the summationsof forcesand moments on each of the masses n the devices The lineardrivestructure had the ollowing coupled equationsof motion.myO 0 0 cy0 0 0[ 0T;:]V [O I & 0... ,r]Yo c a [ K, ]V = O (1)

where

and q sthemass of the ith mass, Im3 is the mass mome...of inertia ofthe semaphore mass, J3 is the area moment of inertia of the semaphoremass, h is the gap distance between the semaphore mass and thesubstrate, p is the viscosity of air, A, is the horizontal area of the ithmass, and [ I is the stiffnessmatrix for the linear drive structure. xl,x2,~ 3 .nd O3 are defined in Fig. 6. [I ] is defined in the Appendix.different set of equationsof motion:The motion of the angulardrive structure was described by a

m y O 0 0O I M O 00 O m 2 0

0 O I n pwhere

c * o 0 00 O C 2 0

U 0 cl00 00 o c 2

and q s the mass of the ith mass, Jj is the polar moment of inertia ofthe ph mass, A, is the horizontal area of the ith mass, and [Ka] s thestiffness matrix for the angular drive structure. xl, 01, x2, and 8, aredefined in Fig.7. [K,] is defined in the Appendix.

By assuming harmonic response to the free vibration problem,each of the above 2 statically coupled sets of matrix equations can bereduced to an eigen problem and solved for their natural (resonant)frequenaes and their mode shapes.It should be noted that in equations (1) and 5 ) that thestiffness matrices are proportional to Young's modulus times the beamwidth cubed. Since these terms only appeared in the stiffness matrix,the resonant frequency was directly related to their product. Thisimplies that the analytic predictability of the resonant frequenciesdepended strongly on the accuracy to which both beam dimensionsandYoung'smodulus were known. For example, for a 10%change in beamwidth, the predicted fundamental resonant frequency of the structurewould change by about 15 . This also implies that, if the beamwidths were known, the value of Young's modulus can be predicted.EXPERIMENTThe structures were fabricated using the four-mask processdeveloped by [31. One of the advantages of this process was that allthe critical featureswere defined with one mask, reducing errors due tomisalignment. The structural layer defining the oscillating device was

phosphorus-doped polysilicon that had been annealed at 1050C for 30minutes in N p After the final rinse and drying, every structure wasattached to the substrate, but could be freed easily with a probe tip.The resonant frequencies, amplitudes of motion, and modesshapes of several of each of the two types of st ructures weredetermined. Structures ranging in overall length from 398 to 864 pmwere tested; the curved-combdriven structures being the shorter. Allobservations were made visually under 1OOOX magnification both7 i r


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directly through a microscope and on a video monitor. Both continuousand stroboscopic illumination was used. A dc bias voltage was appliedto pad I (Figure 6) which is connected to the ground plane andsemaphorestructure,and to pd 2. The comb ~ o ~ e ~ t e do p 12 neededto be set at the same potential as the semaphore structure so therewould be no electrostatic attraction between the two. A sinusoidaldrive voltage was applied to pad 3, which was connected to the otherset of fixed-electrode fingers.All testing was performed in ambient air. For all measurementsof amplitude, the dc bias voltage was 50 V and sinusoidal drivingvoltage zero-to-peakmplitude was 10 V. The resonant frequencies ofthe semaphore structures were determined by scanning through drivingvoltage frequencies and finding the bands of maximum structureresponse amplitude. The amplitudes of the structure motion weredetermined by observing the difference in position between the blurenvelopeboundary of the resonating structure and the structure at rest.

(drive type)A

RESULTSA total of five multiple-mode resonant structures were testedand compared to computer simulations. f the five, four were lineardrive-base designs, and one was an angular drive-base design.Frequency predictions for all structures and all modes were generallyvery good. Predictions of vibra tion mode shapes were in moderateagreement with experiment, but occasionally, mode amplitudes weretoo small to determine.Accurate simulation of the structureswas dependent on knowingboth the beam width and Young's modulus for the polysilicon. Thefabricated beam width was measured using SEM micrographs. Thedimension between the sides, the planes of which faced the samedirection, of two adjacent beams gave a reference length from whichother dimensionscould be scaled. Thebeams did not have a Fectangularcross-section, but an equivalent beam width, which corresponded to theactual area moment of inert ia of the beam, was used in the simulation.In order to determine an estimate of Young's modulus, the value ofYoung's modulus used in the simulation for one single structure wasvaried until the predicted first resonant frequency of the that s tructurematched the experimental first resonant frequency. The resultingestimate of the value for Young's modulus was 150 GPa; this value wasthe same as that found by others [21 who have used an identicalfabrication process. This value for Young's modulus was usedthroughout the rest of the simulations.

beam length beam kngthtun) W)300 300,

ISecondResonant Mode Shape L

(linear)F(linear)H

[Third ResonantMode Shape

300 200+m 173t

Fig. 8. First three mode shapes of the semaphore portion of amultimode resonant structure. The dark figures represent thesemaphore at rest, while the light figures epresent the semaphore atrrsonance.

expS t r u m or

Table1. St ruaUrr dimemi-structure I Drive-base I Semaphore I

Resonant Frequency EHz1st ZIUlI 3rd(drive type)A(linear)C

(linear) I IC 400 300t

model mode mode modemodel 1705 19,370 28,320exp 165Of125 18,765f225 2spOOf260exp 262Of80 12,720flOO 21,300f150(linear)

E(linear)F

(angular) I* beam ip attachedto edge of semaphore masst beam tip attached to center of semaphore mass

d e l 2613 12,910 21,470model 4050 12,390 27,320exp 4050flOO 12,375f165 28,230f200exD 4275 +lo0 16,100+lo0 32550 f150(linear)H(angular)d e l 4242 15,640 3 1 2 0exp 3635flOO 15,700flOO -model 3796 15,820 115,300

The dimensions of the five structures are shown in Table 1Experimental and simulation results are compared in Table 2. Thedynamic model predicted the mode shapes and resonant frequencies ofthe threeperceptible resonances of the linear drive structures and ofthe two perceptible r es ~n an ~e sf the angular drive structures. Once thevalue of Young's modulus was determined, the predicted resonantfrequencies were all within 5% of the actual resonant frequencies forallmode shapes.Three perceptible resonant frequencies and associated modeshapes were found for each of the linear drive structures and tworesonant frequencies and mode shapes were found for the angular drivestructures. The resonant frequencies are shown in Table2. The modeshapes or the structures are shown in Fig. 8and the numeric valuesareshown in Table 3. The mode shapes for the linear and angular drivestructureswere similar for the first and second modes, but no third modewas perceptible in the angular drive structures. This lack of aperceptible third mode was probably due to the shorter overall lengthof the angular drive structures . The short length simulation resulted inextremely high resonant frequency. A fourth resonant mode waspredicted for the linear drive structures, but the lowest predictedresonant frequency for this mode was 137 kI-h and resonance at thishigh fmquency may have been precluded by the damping .The analytic predictability of the frequency depended stronglyon the accuracy of the knowledge of the beam dimensions, butsimulations have shown that the mode shapes are relativelyinsensitive to variations in the beam width or in the value of Young'smodulus. Similarly, the dynamic model showed that variations in theamount of oiscous damping, assuming fully developed flow under theplate, had little effect on the predicted natural frequencies and modeshapes of the structures.

ome v ry long structures were fabricated. The longest struchve(structure A)maswed 864 pm from the free end of the semaphore massto the attachment of the linear drive-base beams to the substrate.There was no perceptible contact between the semaphore and subs trateduring the testing of this device and the experimental resonantfrequencies were within 3%of the predicted frequencies, implying nosurfaceinteractions or contact. A similar structure that was964p ongwould not resonate and it is assumed that this structure did indeedcontact the substrate.

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Table 3. Simulated and experimental mode shapes of linear Mveatntcture. Simulated values are normalized by the experimentalvalue forX2or by X3 if X2wasnot measurable.

< 10.2A -0 4 fl(linear)

C(linear)

moiel -3 -0.7 2.7model 3.5 -0.3 1model 0.2 3 0.8model -5.5 -1.6 4.6model 5.5 -0.8 1.7

3rd exp 3.5fl -0 3fl1st exp O 3fl 2 f152nd exp -5.5 fl 6 f153rd exp 5 S f l 4 fl

I(angular)H I 1st exp I I 4 + 1 2 fld e l 0.5 4.O 1.5-1.0H.5 4 f l

1 o 2.1odel 1.12 n d -negative values indicate the relative positions of the componentsofthe structuresnot measurable

CONCLUSIONFive multiple degree-of-freedom micromechanical structuresthat can be tuned to respond in a desired vibration mode have beendesigned, fabricated, and tested. The linear and angular vibrationmodes of these structures can be preferentially excited. Further, thelargest amplitude of structure motion can be made to occur at somedistance from the drive. This can be exploited to provide the means forexating resonant structures several hundred micrometers away from theresonatingmass. Theoretical dynamic models have been developed forboth the linear drive and angular drive resonant structures and thesepredict the resonant frequencies to within 5 as well as predict thevibrational mode shapes of the fabricated structures. The modelspredict no mode shape difference and little frequency change as theamount of Viscous damping varies. A value of 150GPa hasbeen derivedfor Young's modulus and it matches the value reported elsewhere [Z]. Aresonant frequencyas ow as1650Hz hasbeen measured for a cantilevertype structure that has an overall length of 864p

REFERENCESH. Guckel, J. J. Sniegowski, T.R. Cristenson, F. Raissi, TheApplication of Fine Grained, Tensile Polysilicon toMechanically Resonant Transducers , Transducers '89,5thInternational Conference on Solid-state Sensors and Actuators,Montmux Switzerland,25-30 une1989.William C. Tang, TuCuong H. Nguyen, Michael W. Judy, andRoger T. Howe, Electrostatic-Comb Drive of Latera lPolysilicon Resonators , Transducers '89,5th InternationalConference on Solid-state Sensors and Actuators, Montreux,Switzerland, 25-30 June1989.William C. Tang, Tu-Cuong H. Nguyen, and Roger T. Howe,Laterally Driven Polysilicon Resonat Microstructures , IEEEMicro Elcetro Mechanical systems Workshop, Salt Lake City,Utah 20-22 ebruary 1989).

A.P. Pisano and Y.-H. Cho, Mechanical Design Issues inLaterally-Driven Microstruct ures , Transducers '89,5thInternational Conference on Solid-State Sensors and Actuators,Montreux, Switzerland,25-30 June1989.

Y.C. Tai and R. S. Muller, Frictional Study of IC-ProcessedMicromotors , Transducers '89,5th International Conference onSolid-state Sensors and Actuators, Montreux, Switzerland, 25-30June1989.K.J. Gabriel, F. Behi, R. Mahadevan, and M.Mehregany, InSitu Friction and Wear Measurements in Integrated PolysiliconMechanisms , Transducers '89,5th International Conference onSolid-state Sensors and Actuators, Montreux, Switzerland, 2530 une1989.M.Mehregany, S.F. Bart, L.S.Tavrow, J.H. Lang, S.D. Senturia,and M.F. Schlecht, A Study of Three Microfabricate Variable-capacitance Motors , presented at Transducers '89,5thInternational Conference on Solid-State Sensors and Actuators,Montnmx, Switzerland,25-30 June1989.

ACKNOWLEDGEMENTThe authors would like to acknowledge the assistance, support,and efforts of Tom Booth as well as the others of the staff of theBerkeley Microfibrication Facility without whom this work would nothave been suamsful.

APPENDIXThe stifmessmatrix [ I for the linear drive is defined as

follows.

0 29346 -2g3Q6 ?($+313b +3bl1 -where 4 4a=? a = ?lgl =11 12 13d 1 = 6 T1 d2'6-i- E 2 d 3 = 6 713

2 3l3 2b

2=-11 12, and I3 are the area moments of inertia of the beams and 11.12, 13,and b am defined in Fig. 9.

Fig.9. Deflnitiomfor dimensions of inear drive structure.1 3

1 - -


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Fig.lo. D a t b n S or dimensionsof angular drive tructure.

where3=11

- S + l 2 + b ) ) Q = 12 s= 4 - s + l 2 + b )

11 and I2are the area momentsof inertia of the beams and 11.12, r, s,andb are definedin Fig.10.

1 4

I