Optimization-of-Passive-Air-Damping-of-MOEMS-Vibration-Senso(3)

442 A. Kainz et al. / Procedia Engineering 87 ( 2014 ) 440 – 443

(a)Plotofthedampingcoe cientsoftheteststructureob-tainedwithOpenFOAMforavariationofthefrontgapxfr.

Forsmallvaluesofxfr,thecontributiondfrbecomesdomi-nant,whereasforlargevaluesofxfr,dCislargest.

(b)Setupforthemeasurementofthetransferfunctionofoursensor.Theactuationviaapiezodrivenshaker[5]isrecordedbyalaserDopplervibrometer(PolytechMSA400)andthetransfercharacteristicsofthesensorbytheopticalread-outwithinthesensormounting.

Figure2:FVMresults(a)andMeasurementsetup(b).

chipandseismicmass.Thedampingduetotheoppositefacedδandfrontfacesdfrofthemassaresmaller,butstillrelevant.FordCanddδcanbemodeledverywell,evenbyexactsolutionsoftheNavier-StokesEquation[2–4].ItcanbeseenthatdCanddδareduetoviscousstressonly,thepressurecontributionvanishes.Withdde nedasforceovervelocityd=F/v=τAm/v,whereτisthestressatthesurfaceAm,theycanbecalculatedto

AmτδAmτCAm

dC==μ,anddδ==(1 i)Am.(3)

vgv2Thus,thereisonlyonerelevantparameterfordδ,namelythebaseareaoftheseismicmassAm.FortheCouette

contribution,therearetworelevantparameters,againAmandtheheightofthegapg.IncontrasttodCanddδ,analyticalmodelingofthefrontcontributionisveryhardand-toourknowledge-thereexistsnoappropriatemodel.Thus,dfr,whichiscausedbypressureforces,iscoveredbytheFVMsimulations.Evenwithoutananalyticalmodel,therelevantparametersforthiscontributioncansafelybeidenti edastheareaofthefrontfaceAfraswellasthewidthofthecorrespondinggapxfr.Employingasqueeze lmdescription,forexample,willseverelyunderestimatethedampingforcesbyseveralordersofmagnitudeforappropriatevaluesofxfr.3.MeasurementsandResults

Theresultsoftheprevioussectioncannowbeusedtodesignastructurewithawell-de neddampingratio.OnemightnotwanttovaryalloftherelevantparametersAm,Afr,g,xfr,sincetheya ectotherquantitiesofthesystem.Inourcase,wetakeAmandAfras xedquantitiesastheyin uencethemassandthustheresonancefrequency.Thesizeofthegapgresultsfromthefabricationprocessandcanbeadjustedtoacertaindegree(fromafewtoafewtensofmicrons),herewehaveg=12.4μm.Thefrontgapxfrcanbechosenmorefreely.Itispossibletosetittoarangefromafewtoafewhundredsofmicrons.Sinceweexpectedlargede ectionsformeasurementsinvacuum,wetookxfr=40μmforoutteststructure.

WeranFVMsimulationsfortheseparametersandothervaluesofxfr.Theresultingdampingcoe cientscanbeviewedinFig.2a.Notethat,whiletheshearstresse ecingdCanddδstemsfromtheviscosity,dfrisduetopressure.Thetotaldampingcoe cientofourteststructurewascalculatedtod=dC+dδ+dfr=3.45mg/s.Dueto nitesizee ects,theanalyticalmodelsslightlyoverestimatethecorrespondingcontributionsresultingind=dC+dδ=3.59mg/s.Sincetherearestillothersmallcontributionsunaccountedfor,weexpectthemeasured