Optimization-of-Passive-Air-Damping-of-MOEMS-Vibration-Senso(2)

A. Kainz et al. / Procedia Engineering 87 ( 2014 ) 4

40 – 443 441

(a)Micrographofourteststructure.Theoscillationsof

theSimassinducealight uxmodulationthroughtheholesintheSimassandtheCrapertureontheglass.TheairdampingdependsonthedistancesfromthemasstotheSiframe(in-plane)andtotheglasswall(out-of-plane)[1

].

(b)Componentscontributingtotheairdampingofthedevice.Couette ow(dC),shearwaves(dδ)aswellaspressureandviscousforces(dfr)actingonthefrontoftheseismicmassarethemajorcontributions.

Figure1:Layout(a)anddampingcomponents(b)oftheMOEMS.

i.e.inevacuateddevices.Duetothestrongin uenceofviscousandpressureinduceddamping,itismandatorytoisolateandcontroltherelevantquantitiesthatdetermined.Thisallowsfortuningthetransferfunction,especiallyheightandwidthoftheresonancepeak,oftheMOEMSto ttheneedsofagivensensingpurpose.Forinstance,avibrationsensorfavorsalowpeaktoextendthemeasurementregimeandtoavoidringing,whereasaresonantsensorrequiresaveryhighpeakforhighsensitivity.

Inordertodescribeandquantifythecontributionstotheairdamping,onecanemployanalyticalmodelsand/ornumericalapproaches.Forthepresentcase,acombinationofbothwasused,sinceontheonehandanalyticalmodelsareverytimesavingande cient,butontheotherhandcompactandplausibleonesareavailableonlyforfewverysimplegeometries.Asnumericalapproach,wechosethe nitevolumemethod(FVM)implementedintheopensourcesoftwareOpenFOAM®.TheFVMisanumericalmethodforsolvingpartialdi erentialequationsandatypicalandpreferablechoiceforcomputational uiddynamics(CFD)problems.ThisismainlyduetothefactthattheFVMbyde nitionsatis esmass,momentumandenergybalance,threeofthemostfundamentalrequirementsfor uiddynamics.

2.GoverningEquationsandDampingContributions

Theoscillationofthemicrostructurecanbedescribedbyadamped(d),driven(K=Kex),harmonicoscillatorwithmassmandsti nessk.AmicrographofateststructureisdepictedinFig.1a.Thecomplexvalued,frequencydependenttransferfunctionA(ω=2πf)ofthesensingdeviceisthusgivenby

Kω2

A(ω)=.+km ω2+iω(1)

Thedampingparameterdresultsfromtheinduced,incompressible owofair(densityρ,dynamicviscosityμ)

betweenstructureandboundingwallsthatsatis estheNavier-StokesEquationandmassbalance

1μ v

+(v· )v= p+Δvand

·v=0,

(2)

respectively.Here,pisthe uid’spressureandvisthe uid’svelocitywhichhastosatisfytheno-slipcondition

v=vstronthemicrostructre.AgraphicoverviewofthecontributionstotheairdampingcanbeseeninFig.1b.TheusuallylargestcontributiondCisduetothesteeplinearvelocitypro leoftheCouette-like owbetweenglass