EE C245 – ME C218Introduction to MEMS DesignFall 2007Fall 2007Prof Clark TC NguyenProf. Clark T.-C. NguyenDept of Electrical Engineering & Computer SciencesDept. of Electrical Engineering & Computer SciencesUniversity of California at BerkeleyBerkeley, CA 94720yLt 19 R F IIEE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 1Lecture 19: Resonance Frequency IILecture Outline• Reading: Senturia, Chpt. 10gp• Lecture Topics:ª Energy Methods(Virtual Work(Virtual Work( Energy Formulations( Tapered Beam Examplepp( Estimating Resonance FrequencyEE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 2The Raleigh-Ritz Method• Equate the maximum potential and maximum kinetic energies:• Rearranging yields for resonance frequency:ω = resonance frequencyW= maximum potential Wmax maximum potential energyρ = density of the structural materialmaterialW = beam widthh = beam thicknessŷ(x) = resonance mode shapeEE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 3ŷ(x) = resonance mode shapeExample: Folded-Beam Resonator• Derive an expression for the resonance frequency of the flddb lfFolded-beam suspensionfolded-beam structure at left.Shuttle w/ mass MsFolding truss w/ mass Mt\2EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 4Anchorh = thicknesstGet Kinetic EnergiesFolded-beam suspensionShuttle w/ mass MsFolding truss w/ mass Mt\2EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 5Anchorh = thicknesstFolded-Beam SuspensionFldi TFolding TrussyxzComb-Driven Folded Beam ActuatorCombDriven Folded Beam ActuatorEE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 6Get Kinetic Energies (cont)Folded-beam suspensionShuttle w/ mass MsFolding truss w/ mass Mt\2EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 7Anchorh = thicknesstGet Kinetic Energies (cont)Folded-beam suspensionShuttle w/ mass MsFolding truss w/ mass Mt\2EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 8Anchorth = thicknessGet Potential Energy & FrequencyFolded-beam suspensionShuttle w/ mass MsFolding truss w/ mass Mt\2EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 9Anchorh = thicknesstBrute Force Methods for Resonance F DtitiFrequency DeterminationEE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 10Basic Concept: Scaling Guitar Strings Mhil RGuitar StringeμMechanical ResonatormplitudLow QHigh QVib. AmVibrating “A”String (110 Hz)110 HzFreq.Performance:L=40 8μm[Bannon 1996]String (110 Hz)Stiffnessfo=8.5MHzQvac=8,000Q~50Lr=40.8μmmr~ 10-13kgWr=8μm, hr=2μmd=1000ÅV=5Vrokf1=Freq. Equation:Qair50d=1000Å, VP=5VPress.=70mTorrEE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 11Guitarromfπ2Freq.MassAnchor LossesFixed-Fixed Beam ResonatorProblem: direct anchoring to the substrateÖanchorElastic WaveRadiationGapAhsubstrate anchor radiation into the substrate Ö lower QGapAnchorAnchorElectrodeSolution: support at motionless nodal points Öisolate resonatorQ = 300 at 70MHzÖisolate resonator from anchors Ö less energy loss Ö higher QSupporting BeamsFree-Free Beam ResonatorLAnchorLrFree-Free BeamAnchorEE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 12Q = 15,000 at 92MHzFreeFree Beam92 MHz Free-Free Beam μResonator hl h • Free-free beam μmechanical resonator with non-intrusive supports Ö reduce anchor dissipation Ö higher QEE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 13Higher Order Modes for Higher Freq.2ndMode Free-Free Beam3rdMode Free Free BeamDistinct Mode Distinct Mode Shapes-60-57B]90135180]Lr= 20.3 Electrodes-66-63mission [d-4504590se [degreeμmAnchor-72-69Transm-180-135-90-45PhasQ Q = 11,500= 11,500h = 2.1 EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 14Support BeamAnchor72101. 31 101. 34 101. 37 101. 40Frequency [MHz]180μmFlexural-Mode Beam Wave EquationuyuTransverse Displacement = maW= widthzLxFdxFF∂+h• Derive the wave equation for transverse vibration:dxxF∂+EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 15Example: Free-Free BeamWzh• Determine the resonance frequency of the beammfqyfm• Specify the lumped parameter mechanical equivalent circuit• Transform to a lumped parameter electrical equivalent pp qcircuit• Start with the flexural-mode beam equation:4422uAEIu∂∂⎟⎟⎠⎞⎜⎜⎝⎛=∂∂EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 1642xAt∂⎟⎠⎜⎝∂ρFree-Free Beam Frequency• Substitute u = u1ejωtinto the wave equation:(1)• This is a 4thorder differential equation with solution:(2)• Boundary Conditions:()EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 17Free-Free Beam Frequency (cont)• Applying B.C.’s, get A=C and B=D, and(3)• Setting the determinant = 0 yields• Which has roots at•Substituting (2) into (1) finally yields:•Substituting (2) into (1) finally yields:Free-Free Beam F EtiEE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 18Frequency EquationHigher Order Free-Free Beam ModesMore than 10x increaseFundamental Mode (n=1)1stHarmonic (n=2)Harmon c (n )EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 192ndHarmonic (n=3)Mode Shape Expression• The mode shape expression can be obtained by using the fact that A=C and B=D into (2), yielding• Get the amplitude ratio by expanding (3) [the matrix] and solving, which yieldssolv ng, wh ch y elds• Then just substitute the roots for each mode to get the expression for mode shapeFundamental Mode (n=1)EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 20[Substitute ]Lumped Parameter Mechanical Eilt Ci itEquivalent CircuitEE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 21Equivalent Dynamic Mass• Once the mode shape is known, the lumped parameter equivalent circuit can then be specified• Determine the equivalent mass at a specific location x using knowledge of kinetic energy and velocityWLocation xzhMaximum Kinetic EnergyDensityEquivalent Mass =EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 22Maximum Velocity @ location xMaximum Velocity
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