EE245: Introduction to MEMSModule 10: Resonance FrequencyCTN 11/1/09Copyright @2009 Regents of the University of CaliforniaEE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 1EE C245 – ME C218Introduction to MEMS DesignFall 2009Prof. Clark T.-C. NguyenDept. of Electrical Engineering & Computer SciencesUniversity of California at BerkeleyBerkeley, CA 94720Lecture Module 10: Resonance FrequencyEE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 2Lecture Outline• Reading: Senturia, Chpt. 10: §10.5, Chpt. 19• Lecture Topics:ª Estimating Resonance Frequencyª Lumped Mass-Spring Approximationª ADXL-50 Resonance Frequencyª Distributed Mass & Stiffnessª Folded-Beam ResonatorEE245: Introduction to MEMSModule 10: Resonance FrequencyCTN 11/1/09Copyright @2009 Regents of the University of CaliforniaEE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 3EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 3Estimating Resonance FrequencyEE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 4EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 4LrhWrVPviClamped-Clamped Beam μResonatorωωοivoiQ ~10,000viResonator BeamElectrodeio]cos[ tVvoiiω=]cos[ tFfoiiω=Voltage-to-Force Capacitive TransducerSinusoidal Forcing FunctionSinusoidal Excitation• ω ≠ ωo: small amplitude• ω = ωo: maximum amplitude → beam reaches its maximum potential and kinetic energiesEE245: Introduction to MEMSModule 10: Resonance FrequencyCTN 11/1/09Copyright @2009 Regents of the University of CaliforniaEE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 5EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 5Estimating Resonance Frequency• Assume simple harmonic motion:• Potential Energy:• Kinetic Energy: EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 6EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 6Estimating Resonance Frequency (cont)• Energy must be conserved:ª Potential Energy + Kinetic Energy = Total Energyª Must be true at every point on the mechanical structure• Solving, we obtain forresonance frequency:Maximum Potential EnergyMaximum Kinetic EnergyStiffnessDisplacement AmplitudeMassRadian FrequencyOccurs at peak displacementOccurs when the beam moves through zero displacementEE245: Introduction to MEMSModule 10: Resonance FrequencyCTN 11/1/09Copyright @2009 Regents of the University of CaliforniaEE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 7EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 7Example: ADXL-50• The proof mass of the ADXL-50 is many times larger than the effective mass of its suspension beamsª Can ignore the mass of the suspension beams (which greatly simplifies the analysis)• Suspension Beam: L = 260 μm, h = 2.3 μm, W = 2 μmSuspension Beam in TensionProof MassSense FingerEE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 8EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 8Lumped Spring-Mass Approximation• Mass is dominated by the proof massª 60% of mass from sense fingersª Mass = M = 162 ng (nano-grams)• Suspension: four tensioned beamsª Include both bending and stretching terms [A.P. Pisano, BSAC Inertial Sensor Short Courses, 1995-1998]EE245: Introduction to MEMSModule 10: Resonance FrequencyCTN 11/1/09Copyright @2009 Regents of the University of CaliforniaEE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 9EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 9ADXL-50 Suspension Model• Bending contribution:• Stretching contribution:• Total spring constant: addbending to stretchingEE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 10EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 10ADXL-50 Resonance Frequency• Using a lumped mass-spring approximation:• On the ADXL-50 Data Sheet: fo= 24 kHzª Why the 10% difference?ª Well, it’s approximate … plus …ª Above analysis does not include the frequency-pulling effect of the DC bias voltage across the plate sense fingers and stationary sense fingers … something we’ll cover later on …EE245: Introduction to MEMSModule 10: Resonance FrequencyCTN 11/1/09Copyright @2009 Regents of the University of CaliforniaEE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 11EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 11Distributed Mechanical Structures• Vibrating structure displacement function:• Procedure for determining resonance frequency:ª 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, …)ª Determine the maximum kinetic energy when the beam is at zero displacement (e.g., when it experiences its maximum velocity)ª Equate energies and solve for frequencyŷ(x)Maximum displacement function (i.e., mode shape function) Seen when velocity y(x,t) = 0 EE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 12EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 12Maximum Kinetic Energy• Displacement:• Velocity:• At times t = π/(2ω), 3π/(2ω), …ª The 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):]sin[)(ˆ),(),( txyttxytxvωω−=∂∂=]cos[)(ˆ),( txytxyω=0),(=txyVelocity topographical mapping)(ˆ))2()12(,( xynxvωωπ−=+EE245: Introduction to MEMSModule 10: Resonance FrequencyCTN 11/1/09Copyright @2009 Regents of the University of CaliforniaEE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 13EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 13Maximum Kinetic Energy (cont)• At times t = π/(2ω), 3π/(2ω), …• Maximum kinetic energy:0),(=txy)(ˆ))2()12(,( xynxvωωπ−=+Velocity:2)],([21txvdmdK ⋅⋅=)( dxWhdm⋅=ρEE C245: Introduction to MEMS Design LecM 10 C. Nguyen 11/4/08 14EE C245: Introduction to MEMS Design LecM 9 C. Nguyen 9/28/07 14The Raleigh-Ritz Method• Equate the maximum potential and maximum kinetic energies:• Rearranging yields for resonance frequency:ω = resonance frequencyWmax= maximum potential energyρ = density of the structural materialW = beam widthh = beam thicknessŷ(x) = resonance mode shapeEE245: Introduction to MEMSModule 10: Resonance FrequencyCTN 11/1/09Copyright @2009 Regents of the University of CaliforniaEE
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