1EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 1EE C245 – ME C218Introduction to MEMS DesignFall 2007Prof. Clark T.-C. NguyenDept. of Electrical Engineering & Computer SciencesUniversity of California at BerkeleyBerkeley, CA 94720Lecture 17:Energy Methods IIEE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 2Announcements• Midterm Exam Date: ª Tuesday, Oct. 30 (day before Halloween)2EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 3Lecture Outline• Reading: Senturia Chpt. 10• Lecture Topics:ª Energy Methods( Virtual Work( Energy Formulations( Tapered Beam Example( Doubly Clamped Beam Example( Large Deflection Analysis( Estimating Resonance FrequencyEE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 4Comparison With Finite Element Simulation• Below: ANSYS finite element model withL = 500 μmWbase= 20 μm E = 170 GPah = 2 μmWtip= 10 μm• Result: (from static analysis)ª k = 0.471 μN/m• This matches the result from energy minimization to 3 significant figures3EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 5Need a Better Approximation• Add more terms to the polynomial• Add other strain energy terms:ª Shear: more significant as the beam gets shorterª Axial: more significant as deflections become larger• Both of the above remedies make the math more complex, so encourage the use of math software, such as Mathematica, Matlab, or Maple• Finite element analysis is really just energy minimization• If this is the case, then why ever use energy minimization analytically (i.e., by hand)?ª Analytical expressions, even approximate ones, give insight into parameter dependencies that FEA cannotª Can compare the importance of different termsª Should use in tandem with FEA for designEE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 6Large Deflections• Springs often stiffen for large deflections:• Example: center-loaded clamped-clamped beam• Senturia covers this in §10.4, pp. 249-253, so we’ll just give the highlights, herex=0xy4EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 7Apply the Principle of Virtual Work• Guess the deflectionª Good approach: just use the exact solution for small deflections (perhaps obtained using Euler theory)• Include additional energy for large deflectionsª E.g., “self-generated” axial strain due to stretching of the neutral axisEE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 8Potential Energy Function• Total strain in beam:• Strain energy:• Potential energy function:5EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 9Amplitude Stiffened Spring• Find value of c that minimizes U:• Force as a function of tip displacement:• Spring rate = k(c) = F/c:Bending term: more important for thicker beamsStretching term: more important for thinner beamsNonlinearEE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 10Previous: Small Angle Beam BendingNeutral axis of a bent cantilever beamxzy6EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 11Large Angle Beam BendingNeutral axis of a bent cantilever beamxzyEE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 12Folded Suspensions in Large Deflection• Fit cantilever solution to 3rdorder polynomial in deflection• Result: [M.W. Judy, Ph.D. Thesis, EECS, UC Berkeley, 1994]7EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 13Estimating Resonance FrequencyEE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 14LrhWrVPviClamped-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 energies8EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 15Estimating Resonance Frequency• Assume simple harmonic motion:• Potential Energy:• Kinetic Energy: EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 16Estimating Resonance Frequency (cont)• Energy must be conserved:ª Potential Energy = Kinetic Energyª Must be true at every point on the mechanical structureª At resonance, potential and kinetic energies are at their maximum values• Solving, we obtain forresonance frequency:Maximum Potential EnergyMaximum Kinetic EnergyStiffnessDisplacement AmplitudeMassRadian Frequency9EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 17Example: Micromechanical Accelerometer• The MEMS Advantage:ª >30X size reduction for accelerometer mechanical elementª allows integration with IC’sxoxaAccelerationInertial ForceSpringProof MassBasic Operation Principle400 μmAnalog Devices ADXL 78DisplacementmaFxi=∝Tiny mass means small output Ö need integrated transistor circuits to compensateTiny mass means small output Ö need integrated transistor circuits to compensateEE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 18Example: 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 Finger10EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 19Lumped 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]EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 20ADXL-50 Suspension Model• Bending contribution:• Stretching contribution:• Total spring constant: addbending to stretching11EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 21ADXL-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 fingersEE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 22Distributed Mechanical Structures• Vibrating
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