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 18 R FEE C245: Introduction to MEMS Design Lecture 18 C. Nguyen 10/30/08 1Lecture 18: Resonance FrequencyLecture 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 18 C. Nguyen 10/30/08 2Estimating Resonance FrequencyEstimating Resonance FrequencyEE C245: Introduction to MEMS Design Lecture 18 C. Nguyen 10/30/08 3Clamped-Clamped Beam μResonatorviLrhWriQ ~10,000Resonator BeamivoiioVPviωωοElectrodeioviVoltage-to-Force Capacitive Sinusoidal Forcing FunctionSinusoidal Eitti]cos[tVviiω=]cos[tFfiiω=pTransducerForcing FunctionExcitation]cos[tVvoiiω]cos[tFfoiiω• ω ≠ ωo: small amplitude• i litd b h it i EE C245: Introduction to MEMS Design Lecture 18 C. Nguyen 10/30/08 4•ω = ωo: maximum amplitude →beam reaches its maximum potential and kinetic energiesEstimating Resonance Frequency• Assume simple harmonic motion:• Potential Energy:•Kinetic Energy: •Kinetic Energy: EE C245: Introduction to MEMS Design Lecture 18 C. Nguyen 10/30/08 5Estimating Resonance Frequency (cont)• Energy must be conserved:ª Potential Energy + Kinetic Energy = Total EnergyªM b i h hil ªMust be true at every point on the mechanical structureOccurs at peak displacementOccurs when the beam moves through zero displacementdisplacementthrough zero displacementMaximum Potential Maximum StiffnessRadian EnergyKinetic EnergyStiffnessDisplacement AmplitudeMassRadian Frequency• Solving, we obtain forresonance frequency:EE C245: Introduction to MEMS Design Lecture 18 C. Nguyen 10/30/08 6resonance frequencyExample: ADXL-50• The proof mass of the ADXL-50 is many times larger than the effective mass of its suspension beamsªCan ignore the mass of the suspension beams (which ªCan ignore the mass of the suspension beams (which greatly simplifies the analysis)• Suspension Beam: L = 260 μm, h = 2.3 μm, W = 2 μmpμ,μ,μProof MassSuspension Beam Sense FingerEE C245: Introduction to MEMS Design Lecture 18 C. Nguyen 10/30/08 7Suspension Beam in TensionLumped Spring-Mass Approximation• Mass is dominated by the proof massª 60% of mass from sense fingersªM M 162 ()ªMass = M = 162 ng (nano-grams)• Suspension: four tensioned beamsªInclude both bending and stretching terms [A P Pisano ªInclude both bending and stretching terms [A.P. Pisano, BSAC Inertial Sensor Short Courses, 1995-1998]EE C245: Introduction to MEMS Design Lecture 18 C. Nguyen 10/30/08 8ADXL-50 Suspension Model• Bending contribution:• Stretching contribution:•Total spring constant: addbending to stretching•Total spring constant: addbending to stretchingEE C245: Introduction to MEMS Design Lecture 18 C. Nguyen 10/30/08 9ADXL-50 Resonance Frequency• Using a lumped mass-spring approximation:• On the ADXL-50 Data Sheet: fo= 24 kHzªWh th 10% diff ?ªWhy the 10% difference?ª Well, it’s approximate … plus …ª Above analysis does not include the frequency-pulling yqypgeffect of the DC bias voltage across the plate sense fingers and stationary sense fingers … something we’ll cover later on cover later on …EE C245: Introduction to MEMS Design Lecture 18 C. Nguyen 10/30/08 10Distributed Mechanical Structures• Vibrating structure displacement function:ŷ(x)Maximum displacement function (i.e., mode shape function) • Procedure for determining resonance frequency:(, p )Seen when velocity y(x,t) = 0 gqyª Use the static displacement of the structure as a trial function and find the strain energy Wmaxat the point of maximum displacement (e.g., when t=0, π/ω, …)mmum p m (.g.,w ,/,)ª Determine the maximum kinetic energy when the beam is at zero displacement (e.g., when it experiences its maximum velocity)EE C245: Introduction to MEMS Design Lecture 18 C. Nguyen 10/30/08 11maximum velocity)ª Equate energies and solve for frequencyMaximum Kinetic Energyˆ• Displacement:ˆ),(txy∂]cos[)(ˆ),(txytxyω=• Velocity:A i /(2) 3/(2) ]sin[)(ˆ),(),(txyttxytxvωω−=∂∂=•At times t = π/(2ω), 3π/(2ω), …0),(=txyªThe displacement of the structure is y(xt) = 0Velocity topographical mappingªThe displacement of the structure is y(x,t) = 0ª The velocity is maximum and all of the energy in the structure is kinetic (since W=0):EE C245: Introduction to MEMS Design Lecture 18 C. Nguyen 10/30/08 12)(ˆ))(,( xynxvωωπ−=Maximum Kinetic Energy (cont)• At times t = π/(2ω), 3π/(2ω), …0),(=txy0),(=txy)(ˆ))(,( xynxvωωπ−=Velocity:12)],([21txvdmdK ⋅⋅=•Maximum kinetic energy:)( dxWhdm⋅=ρ•Maximum kinetic energy:EE C245: Introduction to MEMS Design Lecture 18 C. Nguyen 10/30/08 13The 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 18 C. Nguyen 10/30/08 14ŷ(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 18 C. Nguyen 10/30/08 15Anchorh = thicknesstGet Kinetic EnergiesFolded-beam suspensionShuttle w/ mass MsFolding truss w/ mass Mt\2EE C245: Introduction to MEMS Design Lecture 18 C. Nguyen 10/30/08 16Anchorh = thicknesstFolded-Beam SuspensionFldi TFolding TrussyxzComb-Driven Folded Beam ActuatorCombDriven Folded Beam ActuatorEE C245: Introduction to MEMS Design Lecture 18 C. Nguyen 10/30/08 17Get Kinetic Energies (cont)Folded-beam suspensionShuttle w/ mass MsFolding truss w/ mass Mt\2EE C245: Introduction to MEMS Design Lecture 18 C. Nguyen 10/30/08 18Anchorh = thicknesstGet Kinetic Energies (cont)Folded-beam suspensionShuttle w/ mass MsFolding truss w/ mass Mt\2EE C245: Introduction to MEMS Design Lecture 18 C. Nguyen 10/30/08
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