A novel nonlinear amplitude-modulation gyroscope incorporating internal resonance

0.54 MB·

by  S. Amir Mousavi Lajimi

Proceedingsofthe25thCANCAM
London,Ontario,Canada,May31-June4,2015
7 A NOVELNONLINEAR AMPLITUDE-MODULATION GYROSCOPEINCORPORATING
1 INTERNAL RESONANCE
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J S. AmirMousaviLajimi,Navid Noori,AmrMarzouk, Behraad Bahreyni,Farid Golnaraghi

7 MechatronicSystemsEngineering,Simon FraserUniversity,B.C., Canada V3T0A3
[email protected],[email protected]
]
t
e
d
- ABSTRACT byanexternalsignalproportionaltotherotationrate[1].
s
n
Thedesignsofrotationratesensors(gyroscopes)areevolv-
i We are presenting the design and the preliminary numerical
. ingtoofferhigherresolution,largersensitivity,betterperfor-
s
andexperimentalanalysesoftwomismatchedCoriolisvibra-
c manceanda lowerfabricationcost. Beam-basedgyroscopes
i torygyroscopesincorporatingnonlinearmodalinteraction.A offer a relatively simple structure and have been in-use for
s
y noveldouble-Hdesignincludestwoclamped-clampedbeams a long time [2]. Gyroscopes suffer from the low sensitivity
h and a suspended mass in the middle connected to the base andlowmechanicalscalefactor. Toovercomethesedifficul-
p beamsviafourshortcantilevers.AnotherdesignisaT-shaped
ties, researchers have investigated beam-mass systems both
[
gyro including a primary doubly-clamped beam and a sec-
experimentally and analytically [3, 4]. Furthermore, to op-

1 ondary sense beam. A combination of analytical, finite ele- erate the beam-mass gyroscope using frequency-modulation
v ment, and experimental analyses are employed to study the
schemetheclosed-formequationrelatingtheangularrotation
5 characteristicsofthenonlineargyro.Thedrivemodematches
rate and the modal frequencieshas been obtained [5, 6] and
6
the structure’s second mode, while the sense mode matches
0 toexploitnonlinearitiesforhigherperformance,thenonlinear
thefundamentalmodeofthestructure.Ourpreliminarystudy
0 responsehasbeencharacterized[7].
0 indicatesthatthebandwidthandthesensitivityoftherotation
Toincreasethesensitivityofanamplitude-modulationgyro,
. rate sensor are improved by employing the nonlinear modal
2 thefrequenciesofthesenseanddrivemodesarematched[8].
0 interaction.
However, the bandwidth and other performance parameters
7
areaffectedadverselyandmaybeimprovedbyotherarrange-
1 Keywords: Gyroscope,ratesensor,internalresonance,sen-
: ments of the modes [9]. The other possibility is to use dif-
v sitivity,bandwidth,noise,saturation
ferent mode shapes of a structure and operate based on the
i
X featuresofnonlinearlycoupledsystemstoincreasetheband-
r INTRODUCTION widthand the sensitivity asit has been shownfor resonators
a
[10]. Furthermore,aparametricallyresonatinggyroscopeof-
fers a higher performance [11]. Recently, Golnaraghi et al.
AconventionalCoriolisvibratorygyro(CVG)referstoamicro-
[12] revealed an application of internal resonance in design-
ormacro-mechanicalstructurewheretheCorioliseffecttrans-
inggyroscopeswithlargerbandwidthsandhighersensitivities
fers some energy from a primary oscillation mode to a sec-
[12].
ondary oscillation mode. The system used as a sensor, in-
cludes a primary(drive) mode excited by an externalsource Inthiswork,weproposemethodsforthedesignandanal-
and a secondary(sense) mode activated by the rotation rate- ysisofanamplitude-modulationgyrowhichisexcitedatthe
dependentterms. To operatea CVG in open-loopmodeand frequency of the drive mode being twice the frequency of
amplitude-modulationscheme,theamplitudeofthesensemode the sense mode. The first and second frequencies are inter-
(pickoff)isusedtoidentifytherotationrate[1].Anotheroper- nally coupled because of the structure’s design and offer a
ationmodeforvibratorygyroscopesistheforce-to-rebalance, higher sensitivity and a larger bandwidth in the sensed re-
orclosed-loop,modewherethesenseamplitudeisestimated sponse.ThefiniteelementanalysisisperformedinAnsysMe-
chanical APDL 14.5 package using static, modal, harmonic, theequationofmotionintheform
and transient analyses. The initial designs are implemented
inCoventorWarepackageandfabricatedusingaSOIMUMPs M r¨1 + C r˙1 + K K1 = Fr+Fe (1)
process [13]. Two designs are used to study two-to-one in- (θ¨2) (θ˙2) (cid:26)K2(cid:27) (cid:26) Fθ (cid:27)
ternally coupled gyroswhere the first one, a double-Hgyro, (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
is a novel design and the second one, a T-shaped structure, where
was previouslystudied as a resonator. A macro-scale model
oftheT-shapedstructureisbuiltandtestedonaratetableand M = M1+M2 −M2r2sin(θ2)
thefrequency-responsecurvesareobtained. Usingalumped- (cid:26)−M2r2sin(θ2) M2r22 (cid:27)
massmodel,amathematicalmodelofthesystemisdeveloped (cid:2) (cid:3)
andsolvednumericallytoobtainthesaturationcurves.
(cid:2)C(cid:3)=(L21c0−1r12 c02), (cid:2)K(cid:3)=Lar1crsin01(−LrLr111221) θ02
DESIGNSandMETHODS
 
The schematic of the two micro-gyro designs with various Fr =M2r2 θ˙22+Ω2 cos(θ2)+2M2r2Ωcos(θ2)θ˙2
configurationsofthedriveandsense electrodesareprovided +(M(cid:16)+M )r(cid:17)Ω2
1 2 1
inFigs. 1and2. InFig. 1,excitingthebasebeamsalongY-
axis,theCorioliseffectinducesaresponseintheX-direction Fe =aesin(fet)
forasingle-axisgyrorotatingabouttheZ-axis(out-of-plane). Fθ =−M2r2r1Ω2sin(θ2)−2M2r2Ωcos(θ2)r˙1
It is worth mentioning that the same structure may be used
for in-plane rate measurements placing the drive electrodes whereparametersandvariablesareintroducedinFig. 4. The
alongtheZ-axisunderthedrivebeamontheSisubstrate.The mathematicalmodel,Eq.(1),isusedtonumericallystudythe
double-Hdesignincludesamasstofurtheramplifythesense behaviourofthesystemandtoderiveareduced-ordermodel
signal. Theseconddesignis a T-shapedstructurepreviously basedonatwo-variableperturbationmethod[14].
usedasaresonatorin[10],seeFig. 2.
Both designs are initially analyzed using MEMS CAD RESULTS
softwareCoventorWareandANSYS14.5Workbenchandfab-
ricatedusingastandardSOIMUMPsprocess. Scanningelec- An Ansys MechanicalAPDL codeis developedto study the
tron microscope(SEM) images of the fabricated devices are static, modal,harmonic,andtransientresponsesofthestruc-
presented in Figs. 1(a) and 2(a) obtained using a FEI Nova tures. To performthe finite elementanalysisof the microT-
NanoSEM 430 SEM System. The structures are designed shapedgyroinareasonableamountoftime,thequalityfactor
suchthatthemodalfrequencyofthesecondmodeistwicethe is set to 100 and a two-dimensional model of the structure,
modalfrequencyofthefirstmodeinteractingwitheachother Fig. 2(a),isstudiedinAnsysMechanicalusingPlane1822D
throughnonlinearmodalinteraction(internalresonance).The structuralsolidelements. ForaT-shapedgyrowithaprimary
pair of sense electrodes provide the required means for dif- 388µm long clamped-clamped beam with width 20µm and
ferentialsensingofthesensedisplacement. Forthedouble-H a secondary 145.5µm long beam with width 8µm, the time-
gyro,Fig. 1,thedriveelectrodesoperatein180degreesphase histories of the drive and sense displacements are computed
differencetoincreasethestrengthofactuation.Properinstru- using transient analysis and shown in Fig. 5. The energy
mentationoftheexperimentationsareshowninFigs.1(b)and istransferredfromthedrivebeam(mode)to the sense beam
2(b).
Toproperlyidentifythenonlineardynamicsofthesystem, (a) (b)
amacro-scalemodeloftheT-shapedsensorisbuiltandtested
on a rate table, see Fig. 3. The parametersof the gyroscope
include a base doubly-clamped beam of length 160.20mm,
thickness 0.47mm, and width 15mm, and a secondary beam
oflength80mm,thickness0.203mm,andwidth15mm. The
actuation and sensing mechanismsinclude two pairs of PSI-
5H4E piezoelectric ceramic transducers and a pair of LTS
15/03 and 15/12 laser displacement sensors. The device is
mountedonanIdealAerosmith1621-200A-TLratetableand
operatedandcontrolledthroughacustomizedMATLABSimulink
codeandadSpacedataacquisitionsystem. Figure1:(a)ASEMimageofthefabricateddouble-Hshaped
A simplified mathematicalmodelbased on lumped-mass gyroand(b)theschematicofthegyroscopeandthetestsetup.
model,Fig.4,oftheT-shapeddesign,Fig.2,isusedtoderive
(a) (b)
Figure 2: (a) A SEM image of the fabricated T-shaped gyro Figure5: Displacementtime historiesof the driveandsense
and(b)theschematicofgyroscopeandthetestsetup. beamsformicroT-gyro(Ω=10◦/s).
Figure6: Fast Fouriertransforms(FFT) of the displacement
timehistoriesformicroT-gyro(Ω=10◦/s).
Figure3: ExperimentaltestsetupfortheT-shapedgyro. The
structureisplacedonthetableinsidetheratetablechamber.
responsecurvesofferingahigherstabilityandalessersensi-
tivitytowherethemeasurementsaretakenalongthefrequency-
responsecurve.
Anumericalsolutionoftheequationofmotion,Eq. 1,is
computedforanincreasingamplitudeof excitationbydirect
integration and plotted in Fig. 8. The sense mode starts to
grow after a certain threshold of the excitation amplitude is
passed where the drive amplitude reaches to a limit and re-
mainsunchanged. Thesaturationphenomenaappearsinsys-
Figure4:Alumped-massmodeloftheT-shapedgyroscope.
tems having internally coupled modes of vibration. Similar
resultsareobtainedusingatwo-variableperturbationmethod
andexperimentalanalysis.
(mode)throughinternalresonanceandtheCorioliseffectgiv-
ing rise to the large amplitude of the sense beam. A fast CONCLUSIONS
Fouriertransform(FFT)ofthesenseanddrivetime-histories
aregiveninFig. 6indicatingthefrequencycomponentsofthe The coupled-systemresults in a multi-frequencyresponse in
time-responses.Theexcitationfrequency(thesecondpeak)in the sense direction of the proposed novel Coriolis vibratory
thedrivedirectiontransformsintotheprimaryfrequency(the gyros. Initial results show improvements in the sensitivity
firstpeak)inthesensedirectionthroughtheinternalresonance and the bandwidth of the open-loop gyroscope. Using in-
andtheCorioliseffect. ternal resonance offers a method to overcome the complexi-
InFig. 7,theexperimentalfrequency-responsecurvesfor tiesofaclosed-loopgyro. Employingthetwo-to-oneinternal
macro-structureofFig. 3arepresented.Repeatingtheexperi- resonancephenomenon,awidebandwidthhighamplitudere-
mentforasetofexcitationamplitudes,afamilyoffrequency- sponseinthesensedirectionisachievedimprovingthestabil-
responsecurvesareobtainedshowinganincreaseintheband- ityandthesensitivityofthedevice. Furthermore,separating
width and the sense amplitude. For small excitation ampli- the driveand sense mode frequencies(for examplethe drive
tudes, the response is similar to linear systems, however for frequency is twice bigger than the sense frequency-i.e a 2:1
largeexcitationamplitudesaflatregionappearsinthefrequency- internalresonance),thesensedsignalcouldbefilteredforfre-
[4] Lajimi, S. A. M., Abdel-Rahman, E., and Heppler,
G., 2014. “Modeling and sensitivity analysis of a
memsvibratoryrotationratesensor”. InProceedingsof
SPIE 2014,SensorsandSmartStructuresTechnologies
for Civil, Mechanical, and Aerospace Systems 2014,
Vol.9061,pp.90612I–90612I–11.
[5] Lajimi, S. A. M., Heppler, G. R., and Abdel-Rahman,
E., 2014. “The application of a new beam-rigid body
mems gyroscope in the frequency-modulation mode”.
In Nano/Micro Engineered and Molecular Systems
(NEMS), 2014 9th IEEE International Conference on,
Figure 7: Experimental frequency-response curves of the
pp.586–591.
sensebeamformacroT-gyroshowinganincreaseintheband-
widthoftheresponse.
[6] Lajimi, S., Heppler, G., and Abdel-Rahman, E., 2014.
“A parametric study of the response of a beam-
rigid-body microgyroscope”. In ASME 2014 Inter-
national Mechanical Engineering Congress and Ex-
position, American Society of Mechanical Engineers,
pp.V010T13A052–V010T13A052.
[7] Lajimi, S., Heppler, G., and Abdel-Rahman, E., 2014.
“Nonlinear dynamics of a beam-rigid body microgy-
roscope”. In ASME 2014 International Design Engi-
neering Technical Conferences and Computers and In-
formation in Engineering Conference, American So-
ciety of Mechanical Engineers, pp. V008T11A036–
V008T11A036.
Figure 8: Numerical force-response curves of the sense and
[8] Zaman, M. F., Sharma, A., Hao, Z., and Ayazi, F.,
drive modesfor macro T-gyro showingthe nonlinearsatura-
2008. “A mode-matchedsilicon-yawtuning-forkgyro-
tionofthedriveamplitude.
scope with subdegree-per-hour allan deviation bias in-
stability”. MicroelectromechanicalSystems,Journalof,
17(6),pp.1526–1536.
quencies up to near the frequency of the drive mode reduc-
ingthenoiseeffectsaroundthenaturalfrequencyofthesense [9] Acar, C., and Shkel, A., 2008. MEMS vibratory gy-
mode.Itisexpectedtoseeaconsiderableimprovementinthe roscopes: structuralapproachestoimproverobustness.
longtermstabilityofthesensorbyreducingtheeffectsofthe Springer.
electronicnoiseonthereadoutcircuit. Furtherinvestigations
[10] Vyas, A., Peroulis, D., and Bajaj, A. K., 2009. “A mi-
are underway to properly characterize the system response,
croresonatordesignbasedonnonlinear1:2internalres-
examine the possibility of using the method to improve the
onance in flexural structural modes”. Microelectrome-
performanceoftheamplitude-modulationgyroscopes,andto
chanicalSystems,Journalof, 18(3),pp.744–762.
examinethepossibilityofusinga closed-looptocontroland
managetheeffectsofnonlinearities. [11] Oropeza-Ramos, L. A., Burgner, C. B., and Turner,
K.L.,2009.“Robustmicro-ratesensoractuatedbypara-
metric resonance”. SensorsandActuatorsA: Physical,
REFERENCES
152(1),pp.80–87.
[1] Curey,R.,Ash,M.,Thielman,L.,andBarker,C.,2004. [12] Golnaraghi,F.,Bahreyni,B.,Marzouk,A.,Sarrafan,A.,
“Proposed ieee inertial systems terminology standard Lajimi,S.A.M.,Pouyanfar,O.,andNoori,N.Vibratory
and other inertial sensor standards”. In Position Loca- gyroscope utilizing nonlinear modal interaction. Filed
tionandNavigationSymposium,pp.83–90. onMay8,2015,ApplicationNumber:62159054.
[2] O’Connor, J. M., and Shupe, D. M., 1983. Vibrating [13] Cowen, A., Hames, G., Monk, D., Wilcenski, S., and
beamrotatingsensor. U.S.Patent4381672. Hardy,B. “Soimumpsdesignhandbook”.
[3] Yang, H., Bao, M., Yin, H., and Shen, S., 2002. “A [14] Golnaraghi, M., Moon, F., and Rand, R., 1989. “Res-
novelbulkmicromachinedgyroscopebasedonarectan- onance in a high-speed flexible-arm robot”. Dynamics
gular beam-mass structure”. Sensors and Actuators A: andStabilityofSystems, 4(3-4),pp.169–188.
Physical, 96(2),pp.145–151.