MEMS技术 第四讲 机械零件原理(1).ppt

MEMS和微系统设计,课程内容,MEMS概述及MEMS设计的概述工艺简要回顾系统设计、工艺设计及版图设计主要的机械、电子元件及其设计基础多域耦合设计：以机电耦合为例子器件性能的估计简单的其他域的元件及其简要设计要点设计实例,第4讲主要内容,1、弹簧设计原理及计算例子2、薄膜设计原理及计算例子3、电容设计原理及计算例子4、电阻设计原理及计算例子5、压电模型,弹簧设计原理及计算例子,1、基本知识2、梁的静态分析3、二阶系统4、梁的动态分析与谐振,YieldStrength(cont.),Below:typicalstressvs.straincurvesforbrittle(e.g.,Si)andductile(e.g.steel)materials,YoungsModulusandUsefulStrength,Storedmechanicalenergy,弹簧的定义,F=KX,ACantileverBeam,预备知识,Whathappen?,DefineStrainisfoundtobeproportionaltostress(for“small”stressesat“lower”temperatures):,possionsRatio,V=PossionsRatio,Possionstrainsfromoffx-axisstress,Forces|tosurfacesSheerStress,Note:compensatingforcesareappliedtotheverticalfacestoavoidanettorque!,=sheerstress,G=sheermodulus,YoungsModulusin(001)Plane,VolumeChangeforaUniaxialStress,Stressesactingonadifferentialvolumeelement,TheresultingchangeinvolumeV,IsotropicElasticityin3D,Isotropic=sameinalldirectionsThecompletestress-strainrelationsforanisotropicelasticsolidin3D:(i.e.,ageneralizedHookesLaw),Basically,addinoff-axisstrainsfromNormalstressesinotherdirections,ImportantCase:planestress,Commoncase:verythinfilmcoatingathin,relativelyrigidsubstrate(e.g.,asiliconwafer),Atregionsmorethan3thicknessesfromedges,thetopsurfaceisstress-freeGettwocomponentsofin-planestress:,ImportantCase:planestress(cont.),Symmetryinthexy-planeThus,thein-planestraincomponentsare:where,where,LinearThermalExpansion,Astemperatureincreases,mostsolidsexpandinvolumeDefinition:linearthermalexpansioncoefficient,Linearthermalexpansioncoefficient,Remarks:valuestendtobeinthe10-6to10-7rangeCancapturethe10-6byusingdimensionsofstrian/k,where10-6K-1=1strian/kIn3D,getvolumethermalexpansioncoefficientFormoderatetemperatureexcursions,canbetreatedasaconstantofthematerial,butinactuality,itisafunctionoftemperature,TAsaFunctionofTemperature,Madou,FundamentalsofMicrofabrication,CRCpress,1998,Thin-FilmThermalStress,Assumefilmisdepositedstress-freeatatemperatureTr,thenthewholethingiscooledtoroomtemperatureTrSubstratemuchthickerthanthinfilmsubstratedictatestheamountofcontractionforbothitandthethinfilm,StressMeasurementViaWaferCurvature,CompressivelystressedfilmbendsawaferintoaconvexshapeTensilestressedfilmbendsawaferintoaconcaveshapeCanopticallymeasurethedeflectionofthewaferbeforeandafterthefilmisdepositedDeterminetheradiusofcurvatureR,thenapply:,LinearThermalExpansion,Butthefilmisattachedtothesubstrate,sotheactualstainisthefilmisthesameasthatinthesubstrate:,弹簧设计原理及计算例子,1、基本知识2、梁的静态分析3、二阶系统4、梁的动态分析与谐振,ReactionForcesandMoments,SignConventionsforMoments&ShearForces,(+)momentleadstodeformationwitha(+)radiusofcurvature(i.e.,upwards),(-)momentleadstodeformationwitha(-)radiusofcurvature(i.e.,downwards),(+)shearforcesproduceclockwiserotation,(-)shearforcesproducecounter-clockwiserotation,BeamSegmentinPureBending,Smallsectionofabeambentinresponsetoatranverseload,Considerasegmentboundedbythedashedlinesdefinedbyd:Atz=0:(i.e.,oftheneutralaxis):segmentlength=dx=Rd(1)Atanyz:segmentlength=dL=(R-z)d(2)Combining(1)&(2):,BeamSegmentinPureBending(cont.),Whyaminussign?SeeSenturia,pp.208-210,Radiusofcurvaturegeometricconnectiontostrain,resultfrombasiccalculus,Combiningthecurvatureandmomentresults:,and,剪切力忽略的条件,Complianceissumofbendingandshear-stresscontributions:,Solvingfor:,Transversaldeflectionofabeamwithaloadattheend,with:,(187,1),(187,2),I:=AreamomentofinertiaofthebeamB:=StrainatthesurfaceofthebeamF:=ForceactingattheendofthebeamdB:=Thicknessofbeamw:=DeflectionofbeambB:=WidthofbeamLB:=LengthofbeamEB:=Youngsmodulusofbeam,187,*W.Beitz,K.-H.Grote,Dubbel,TaschenbuchfrdenMaschinenbau”,*,(187,4),(187,5),Ariamomentumofinertia,(188,1),(188,2),Rectangular:,(188,3),I:=AreamomentofinertiaofthebeamEB:=YoungsmodulusofthebeamFB:=Elasticforceofthebeamatitsendw:=DeflectionofthebeamdB,bB,LB,RB:=Thickness,width,length,andradiusofthebeam,respectively,188,Moreariamomentumsofinertiaarefoundinbookslike:W.Beitz,K.-H.Grote,Dubbel,TaschenbuchfrdenMaschinenbau”orR.D.Blevins,FormulasforNaturalFrequencyandModeShape“,Krieger,Malaba,FL(1987),bB,dB,(187,3),Circular:,RB,Trapezoidshaped:,dB,e,bB,1,bB,2,(188,4),with:,(188,5),Strainonthesurfaceofabeamloadedatitsend,(189,1),Rectangularbeam:,(187,3):,(189,2),189,*W.Beitz,K.-H.Grote,Dubbel,TaschenbuchfrdenMaschinenbau”,*,I:=AreamomentofinertiaofthebeamEB:=YoungsmodulusofthebeamF:=ForceactingattheendofthebeamB:=Strainatthesurfaceofthebeamw:=DeflectionofthebeamdB,bB,LB:Thickness,width,andlengthofthebeam,respectively,Rectangularbeamclampedononesideandloadedattheotherend,xm,wm,(187,1),(187,3):,LB=800mbB=40mdB=20mEB=140GPaF=1mN,190,EB:=YoungsmodulusofthebeamB:=StrainatthesurfaceofthebeamF:=ForceactingattheendofthebeamdB,bB,LB,w:=Thickness,width,length,anddeflectionofthebeam,respectively,Transversaldeflectionofabeamloadedattheend,Thestrainatthesurfaceofabeamclampedatoneendandloadedattheotherendintransversaldirectionislargestatthefixedend.,Strainanddeflectionarenotafunctionsofaninitialstressofthebeam.,Rectangularbeam:,(189,2):,Strainanddeflectionareproportionaltotheforce(linearcharacteristiccurve).,(187,1),(187,3):,191,EB:=YoungsmodulusofthebeamB:=StrainatthesurfaceofthebeamF:=ForceactingattheendofthebeamdB,bB,LB,w:=Thickness,width,length,anddeflectionofthebeam,respectively,Transversaldeflectionofabeamloadedattheend,Rectangularbeam:,Becauseofthetransversestrainthebeamgetsnarroweronthesidewithtensilestressandwiderontheoppositeside.(Withtheexceptionoftheregionnexttotheclamping),Cross-sectionofthebeam:,Withoutload,Withload,bB,bB(1BB),dB,192,(189,2):,EB:=YoungsmodulusofthebeamB:=PoissonsratioofthebeamB:=StrainatthesurfaceofthebeamF:=ForceactingattheendofthebeamdB,bB,LB,w:Thickness,width,length,anddeflectionofthebeam,respectively,Deflectionofbeamsloadedbytheirweight,InthemicroworldtheGoldenGateBridgecouldbeconstructedasasimplebeam.,198,Deflectionofbeamsloadedbytheirweight,199,InthemicroworldtheGoldenGateBridgecouldbeconstructedasasimplebeam.,x,z,F,(208,1),(208,4),208,*,*W.Beitz,K.-H.Grote,Dubbel,TaschenbuchfrdenMaschinenbau”,B(z,x):=Strainofbeamw:=DeflectionofbeamF:=ForceactingattheendofthebeamB,max:=MaximumstrainofbeamEB:=YoungsmodulusofthebeamI:=AreamomentumofinertiaofthebeamLB,dB,bB:=Length,thickness,andwidthofbeam,respectively,(208,2):,209,(208,3):,xm,w0m,B(z,x):=Strainofbeamw:=DeflectionofbeamF:=ForceactingattheendofthebeamB,max:=MaximumstrainofbeamEB:=YoungsmodulusofthebeamI:=AreamomentumofinertiaofthebeamLB,dB,bB:=Length,thickness,andwidthofbeam,respectively,n,D:=Strainofneutralfiberandstressgeneratedbyit,respectivelyI:=Areamomentumofinertiaofthebeamw:=Deflectionofthebeamw0:=DeflectioninthecenterofthebeamF:=ForceactingatthecenterofbeamdB,bB,LB:=Thickness,width,andlengthofthebeamEB:=Youngsmodulusofthebeam,(208,2):,(212,1),Theelasticstrainnofthebeamalongtheneutralfiberneedstobetakenintoaccountalso.Itiscalculatedfromtheextensioninlength.InAnalogyto(90,2)itisassumed:,(212,2),212,90,Effectofthestress,Analogoustotheapproachusedforthemembranes,theequilibriumofforcesattheendsofthebeamiscalculated:,x,z,F,w0,FR,z,FR,(213,1),Anangularbendinglineisusedasanapproximation:,213,90,0,D:=Stress,initialstress,andstressduetostraining,respectivelyI:=Areamomentumofinertiaofthebeamw:=Deflectionofthebeamw0:=DeflectioninthecenterofthebeamF:=ForceactingatthecenterofbeamdB,bB,LB:=Thickness,width,andlengthofthebeamEB:=Youngsmodulusofthebeam,Asanapproximationtheforcesgeneratedbybendingmomentsandstressaresimplyadded:,(214,1),(212,1)+(213,2):,Addingtheforcesisaroughapproximationwhichshallshowtheinterrelationships.MoreexactresultsareobtainedbyFE-calculations.,214,0:=Stressandinitialstress,respectivelyEB:=Youngsmodulusofthebeamw:=Deflectionofthebeamw0:=DeflectioninthecenterofthebeamdB,bB,LB:=Thickness,width,andlengthofthebeamF:=Force,212,(214,1):,LB=800mdB=5mBB=50mEB=70GPa,0=-400MPa,0=+400MPa,w0m,FmN,215,0:=Stressandinitialstress,respectivelyEB:=Youngsmodulusofthebeamw:=Deflectionofthebeamw0:=DeflectioninthecenterofthebeamdB,bB,LB:=Thickness,width,andlengthofthebeamF:=Force,Deflectionofarectangularbeamclampedatbothsides,(214,1):,Ifnoforceisactingonthebeam(F=0),therearethreesolutionsforw0:,w0m,FmN,(216,2),216,k,0:=Criticalstressandinitialstress,respectivelyEB:=Youngsmodulusofthebeamw:=Deflectionofthebeamw0:=DeflectioninthecenterofthebeamdB,bB,LB:=Thickness,width,andlengthofthebeamF:=Forceactingatthecenterofbeam,Deflectionofarectangularbeamclampedatbothsidesasafunctionoftheinitialstress,Ifnoforceisactingonthebeam(F=0),therearethreesolutionsforw0:,w0=0,0k,w0m,(216,1):,(216,2):,(214,1):,217,(216,3):,with:,k,0:=Criticalstressandinitialstress,respectivelyEB:=Youngsmodulusofthebeamw0:=DeflectioninthecenterofthebeamF:=ForceactingatthecenterofbeamdB,bB,LB:=Thickness,width,andlengthofthebeam,Snappingoverofarectangularbeamclampedatbothsides,(218,1),(218,2),wU,wU,218,(216,2):,(214,1):,k,0:=Bucklingstressandinitialstress,respectivelyEB:=YoungsmodulusofthebeamwU:=Deflectionofthebeamatsnappingoverw0:=DeflectioninthecenterofthebeamdB,bB,LB:=Thickness,width,andlengthofthebeamF:=Forceactingatthecenterofbeam,217,Snappingoverofarectangularbeamclampedatbothsides,(218,2):,219,k,0:=Bucklingstressandinitialstress,respectivelyEB:=YoungsmodulusofthebeamwU:=Deflectionofthebeamatsnappingoverw0:=DeflectioninthecenterofthebeamdB,bB,LB:=Thickness,width,andlengthofthebeamF,FU:=Forceandsnappingforce,resp.,Bucklingstress,(216,3):,Theapproximation(216,3)is20%largerthantheexactresult(220,1).,220,207,k,0,EB:=Bucklingstress,initialstress,andYoungsmodulus,respectivelyI:=AreamomentumofinertiaofthebeamF,Fk:=Forceandbucklingload,respectivelyw0:=DeflectioninthecenterofthebeamdB,bB,LB:=Thickness,width,andlengthofthebeam,0,k,GB,EM,EB,M,B:=Initialstress,criticalstress,shearmodulus,Youngsmodulus,andPoissonsratio,respectivelyF,Mt:=ForceandTorquew0,:=DeflectionandangleofrotationdM,dB,LB,bB,RM:=Thickness,length,widthandradiuswU:=Deflectionatsnappingover,Calculationsofdeflectionandforceofcircularmembranesandrectangularbeams,229,ForceF,k,Membrane,TorqueMt,0,3,Beam,*Theeffectofinitialstressisnottakenintoaccount,*,Integrationofbeams,Compliancesadd,弹簧设计原理及计算例子,1、基本知识2、梁的静态分析3、二阶系统4、梁的动态分析与谐振,Infrequencydomain,Dampingcoefficient,transitionfunction,信号与系统的相关知识,为什么要做傅立叶变换：周期函数与非周期函数的傅立叶变换拉普拉斯变换传递函数常用的反变换,频率响应,设阻尼器的阻尼系数为c，则运动方程为,（4-19）,上式中质量块的瞬时位置X(t)有三种情况，取决于阻尼比=c/2m的大小,图4-7,情况1，过阻尼情况,图-9描述了系统的过阻尼情况,质量块的瞬时位置X(t)为:,（4-20a）,一、典型二阶系统的瞬态响应下图所示为稳定的二阶系统的典型结构图。,开环传递函数为:,闭环传递函数为:,这是最常见的一种系统，很多高阶系统也可简化为二阶系统。,称为典型二阶系统的传递函数，称为阻尼系数，称为无阻尼振荡圆频率或自然频率。,特征根为：，注意：当不同时，（极点）有不同的形式，其阶跃响应的形式也不同。它的阶跃响应有振荡和非振荡两种情况。,特征方程为：,当时，特征方程有一对共轭的虚根，称为零(无)阻尼系统，系统的阶跃响应为持续的等幅振荡。,当时，特征方程有一对实部为负的共轭复根，称为欠阻尼系统，系统的阶跃响应为衰减的振荡过程。,当时，特征方程有一对相等的实根，称为临界阻尼系统，系统的阶跃响应为非振荡过程。,当时，特征方程有一对不等的实根，称为过阻尼系统，系统的阶跃响应为非振荡过程。,当输入为单位阶跃函数时，有：,分析：,此时输出将以频率做等幅振荡，所以，称为无阻尼振荡圆频率。,阶跃响应为：,极点的负实部决定了指数衰减的快慢，虚部是振荡频率。称为阻尼振荡圆频率。,阶跃响应函数为：,即特征方程为,特征方程还可为,因此过阻尼二阶系统可以看作两个时间常数不同的惯性环节的串联，其单位阶跃响应为,式中,上述四种情况分别称为二阶无阻尼、欠阻尼、临界阻尼和过阻尼系统。其阻尼系数、特征根、极点分布和单位阶跃响应如下表所示：,可以看出：随着的增加，c(t)将从无衰减的周期运动变为有衰减的正弦运动，当时c(t)呈现单调上升运动(无振荡)。可见反映实际系统的阻尼情况，故称为阻尼系数。,二、典型二阶系统的性能指标及其与系统参数的关系,（一）衰减振荡瞬态过程：,上升时间：根据定义，当时，。,解得：,从上图中，我们可以得出在这种情况下，质量块的振动幅值迅速下降过阻尼适应在易于过量振动的机械和器件（包括微系统）在这种微器件的设计中，设计工程师应该选择一个合适的阻尼器,情况，临界阻尼情况,图-10描述了系统的临界阻尼情况,质量块的瞬时位置X(t)为:,（4-20b）,情况，欠阻尼情况,图-11描述了系统的欠阻尼情况,质量块的瞬时位置X(t)为:,（4-20a）,Damping,SixkindsofdampingAirdamping,EnergyDissipationandResonatorQ,ThermoelasticDamping(TED),Occurswhenheatmovesfromcompressedpartstotensionedpartsheatflux=energyloss,TEDCharacteristicFrequency,GovernedbyResonatordimensionsMaterialpropertiesTable1.materialproperties,QVs.Temperature,MechanismforQincreasewithdecreasingtemperaturethoughttobelinkedtolesshystereticmotionofmaterialdefectslessenergylosspercycle,EvenaluminumachievesexceptionalQsatcryogenictemperatures,DiskResonatorLossMechanisms,airDamping,Manysources:ignoreinternaldampingandacousticradiationfromtheanchorsfocusondragfromthesurroundinggas,GapdimensioncanbeOntheorderofmean-freePathcontinuummodeldoesntapply,Squeeze-FilmDamping,Platesslideinydirection,CouetteDamping,Dragforce=Dampingcoefficient:bEffectiveviscosityatreducedpressure(p50Torr):,Unfixedunits,例题4-12,如图4.25所示，针对空气和硅油做为阻尼流体，估算力平衡微加速度计的阻尼系数。假定微加速度计工作在20,图4-25力平衡微加速度计的阻尼,解:由上图知:梁的质量块宽度，b=510-6m梁的长度，L70010-6m间隙宽度，H=1010-6m从表4.3知:在20时粘度是air18.7510-6Ns/m2si74010-6Ns/m2,Howabouttheresultinasqueezedampingsituation?,OtherParameterin2-ordersystem,ResonantFrequencyQualityfactor,PhysicsmeaningoffandQ,Energymustbeconserved:PotentialEnergy+kineticEnergy=TotalEnergyMustbetrueateverypointonthemechanicalstructure,Solving,weobtainforresonancefrequency:,Example:ADXL-50,TheproofmassoftheADXL-50ismanytimeslargerthantheeffectivemassofitssuspensionbeamsCanignorethemassofthesuspensionbeams(whichgreatlysimplifiestheanalysis)SuspensionBeam:L=260m,h=2.3m,W=2m,Lumpedspring-massapproximation,Massisdominatedbytheproofmass60%ofmassfromsensefingersMass=M=162ng(nano-grams)Suspension:fourtensionedbeamsIncludebothbendingandstretchingtermsA.P.Pisano,BSACInertialSensorShortCourses,1995-1998,ADXL-50Suspensionmodel,Bendingcontribution:Stretchingcontribution:Totalspringconstant:addbendingtostretching,ADXL-50ResonanceFrequency,Usingalumpedmass-springapproximation:OntheADXL-50DataSheet:f0=24kHzWhythe10%difference?Well,itsapproximateplusAboveanalysisdoesnotincludethefrequency-pullingeffectoftheDCbiasvoltageacrosstheplatesensefingersandstationarysensefingerssomethingwellcoverlateron,TounderstandQfromdifferentdomain,Q=2.E/E=m.W0/b=(.W0)/(W1/2.ln2)(参看参考资料中的谐振部分）,QualityFactor(orQ),MeasureofthefrequencyselectivityofatunedcircuitDefinition:,Example:seriesLCRcircuit,Example:parallelLCRcircuit,SelectiveLow-LossFilters:NeedQ,Inresonator-basedfilters:hightankQlowinsertionlossAtright:a0.1%bandwidth,3-resfilter1GHz(simulated)HeavyinsertionlossforresonatorQ1000Example:quartzcrystalresonators(e.g.,inwristwatches)ExtremelyhighQs10000orhigher(Q106possible)mechanicallyvibratesatadistinctfrequencyinathickness-shearmode,弹簧设计原理及计算例子,1、基本知识2、梁的静态分析3、二阶系统4、梁的动态分析与谐振,Vibrationsofbeams,Theresonancefrequencyofasensormaybetheparametertobemeasured.,Thenitneedstobeknown,howtheresonancefrequencyisaffectedbyotherparameters.,Besidesthis,thedistancetootherresonancefrequenciesneedstobeaslargeaspossibletoobtainalargemeasurementrange.,233,Usefulfrequencyrange,f1,Excitingfrequencyf,Amplitudeofthebeam,f2,f3,234,Actuatorsandsensorsusedattheirresonancefrequency,Ifactuatorsandsensorsaredrivenattheirresonance,itsfrequencyneedstobeknownandhowitisinfluenced,because:,235,Wavefunctionofvibratingbeams,LB:=Lengthofthebeami:=Phaseangleofmodeifi:=Resonancefrequencyofmodeiw(x,t):=Wavefunctionui:=PositiondependentpartofawavefunctionofthemodeiAi:=Amplitudeofmodei,LB,w(x),(236,1),(236,2),Thewavefunctionswi(x,t)ofthemodesaresolutionsofadifferentialequation.,236,Resonancesofvibratingbeams,LB:=Lengthofthebeamfi:=ResonancefrequencyofmodeiB:=DensityofbeamAB:=Cross-sectionalareaofthebeamI:=AreamomentumofinertiaofthebeamEB:=Youngsmodulusofthebeami:=Frequencyparameter,(237,1),Theresonancefrequenciesfi(i)ofthemodesareafunctionoftheboundaryconditions(thebearing)ofthebeam.,Inthefollowingtheresonancefrequenciesandwavefunctionsofbeamswillbeintroducedwhichareoftenusedinmicrotechnique.,237,Frequencyparameters,LB:=Lengthofthebeami,i:=Frequencyparameterw(x,t):=Deflectionofthebeam,(239,1),Thefrequencyparametersiandiresultfromtheboundaryconditionthatthe2ndderivativeofthewavefunctionattheendofthebeam(thecurvatureofthebeam)needstobezero.,iisthesolutionoftheequation:,(239,2),iisdefinedby:,239,Resonancefrequenciesofbeamsclampedatoneside,with:,(237,1):,(238,1),LB,w(x),(236,2):,238,LB:=Lengthofthebeamfi:=ResonancefrequencyofmodeiB:=DensityofbeamAB:=Cross-sectionalareaofthebeamI:=AreamomentumofinertiaofthebeamEB:=Youngsmodulusofthebeami,i:=Frequencyparametersw(x,t):=Wavefunctioni:=Phaseangleofmodeiui:=Positiondependentpartofawavefunctionofthemodei,(247,1),(247,2),EB:=YoungsmodulusofthebeamLB:=LengthofthebeamB:=DensityofthebeamI:=Areamomentumofinertiaofthebeamf1:=FundamentalfrequencyofthebeamAi:=Amplitudeofmodeim0:=MassatendofthebeammB:=Masseofthebeam,LB,w(x),(236,2):,247,Bendingvibrationsofabeamclampedatbothsides,with:,(236,2):,(237,1):,Buttheboundaryconditionsaredifferent.,Therefore,theresonancefrequenciesaredifferent:,249,EB:=YoungsmodulusofthebeamLB:=LengthofthebeamB:=DensityofthebeamI:=Areamomentumofinertiaofthebeamfi:=ResonancefrequencyofmodeiAB:=Cross-sectionalareaofbeami:=Phaseangleofmodeii,i:=Frequencyparameters,Excitationofacertainmode,Ifacertainmodeshallbeexcited,itsantinodesneedtobeexcitedwiththecorrespondingfrequency.,Example:The2ndmodeisexcitedelectrostatically.,255,U,Excitationofacertainmode,Example:The2ndmodeisexcitedthermo-mechanically.,256,Resonancefrequenciesofcircularmembranesandrectangularbeams,268,Frequencyf,k,Membrane,Beam,fi:=ResonancefrequencyofmodeiL,b,d:=Length,width,thicknessi:=FrequencyparameterE,:=YoungsmodulusandPoissonsratioI,It:=AreamomentumofinertiaandtorsionalconstantR:=Radius0,k:=InitialandcriticalstressmB,m0,:=Massofbeam,massfixedtobeamanddensity,Beam,Beam,Beam,Distributedsecondordersystemmodeling,TheRaleigh-RitzMethod,Equatethemaximumpotentialandmaximumkineticenergies:Rearrangingyieldsforresonancefrequency:,Example:folded-BeamResonator,Deriveanexpressionfortheresonancefrequencyofthefolded-beamstructureatleft.,UseRayleigh-RitzmethodKineticEnergy:Mustintegratesincethebeamvelocityisafunctionoflocationy!,GetkineticEnergies,Folded-BeamSuspension,GetkineticEnergies,GetkineticEnergies,EquivalentDynamicMass,Forthefolded-beamstructure,wevealreadydeterminedthemaximumkineticenergyAndinourresonancefrequencyanalysis,wevealreadydeterminedexpressionsforvelocity,EquivalentDynamicStoffmess&Damping,Stiffnessthenfollowsdirectlyfromknoledgeofmassandresonancefrequency,AnddampingalsofollowsreadilyfromknowledgeofQorotherlossmeasurands,Withmass,stiffness,anddamping