EE C245 ME C218 Introduction to MEMS Design Fall 2007

Transcription

1 EE C245 ME C218 Introduction to MEMS Design Fall 2007 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA Lecture 17: Energy Methods II EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 1 Announcements Midterm Exam Date: Tuesday, Oct. 30 (day before Halloween) EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 2 1

2 Lecture Outline Reading: Senturia Chpt. 10 Lecture Topics: Energy Methods Virtual Work Energy Formulations Tapered Beam Example Doubly Clamped Beam Example Large Deflection Analysis Estimating Resonance Frequency EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 3 Comparison With Finite Element Simulation Below: ANSYS finite element model with L = 500 μm W base = 20 μm E = 170 GPa h = 2 μm W tip = 10 μm Result: (from static analysis) k = μn/m This matches the result from energy minimization to 3 significant figures EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 4 2

3 Need a Better Approximation Add more terms to the polynomial Add other strain energy terms: Shear: more significant as the beam gets shorter 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)? Analytical expressions, even approximate ones, give insight into parameter dependencies that FEA cannot Can compare the importance of different terms Should use in tandem with FEA for design EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 5 Large Deflections Springs often stiffen for large deflections: Example: center-loaded clamped-clamped beam x=0 y x Senturia covers this in 10.4, pp , so we ll just give the highlights, here EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 6 3

4 Apply the Principle of Virtual Work Guess the deflection Good approach: just use the exact solution for small deflections (perhaps obtained using Euler theory) Include additional energy for large deflections E.g., self-generated axial strain due to stretching of the neutral axis EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 7 Total strain in beam: Potential Energy Function Strain energy: Potential energy function: EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 8 4

5 Amplitude Stiffened Spring Find value of c that minimizes U: Force as a function of tip displacement: Bending term: more important for thicker beams Spring rate = k(c) = F/c: Stretching term: more important for thinner beams Nonlinear EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 9 Previous: Small Angle Beam Bending y z x Neutral axis of a bent cantilever beam EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/

6 Large Angle Beam Bending y z x Neutral axis of a bent cantilever beam EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 11 Folded Suspensions in Large Deflection Fit cantilever solution to 3 rd order polynomial in deflection Result: [M.W. Judy, Ph.D. Thesis, EECS, UC Berkeley, 1994] EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/

7 Estimating Resonance Frequency EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 13 W r v i Clamped-Clamped Beam μresonator Resonator Beam Electrode Sinusoidal Excitation vi L r Voltageto-Force Capacitive Transducer EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 14 V P h i o Sinusoidal Forcing Function Q ~10,000 i v o i vi = Vi cos[ ωot] fi = Fi cos[ ωot] ω ω o : small amplitude ω = ω o : maximum amplitude beam reaches its maximum potential and kinetic energies ω ο ω 7

8 Estimating Resonance Frequency Assume simple harmonic motion: Potential Energy: Kinetic Energy: EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 15 Estimating Resonance Frequency (cont) Energy must be conserved: Potential Energy = Kinetic Energy Must be true at every point on the mechanical structure At resonance, potential and kinetic energies are at their maximum values Maximum Potential Energy Stiffness Displacement Amplitude Maximum Kinetic Energy Mass Radian Frequency Solving, we obtain for resonance frequency: EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/

9 Example: Micromechanical Accelerometer The MEMS Advantage: >30X size reduction for accelerometer mechanical element allows integration with IC s Basic Operation Principle Tiny Tiny mass mass means means small small output output need need integrated integrated transistor transistor circuits circuits to to compensate compensate 400 μm x o x Fi = ma x Displacement Spring a Inertial Force Proof Mass Acceleration Analog Devices ADXL 78 EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 17 Example: ADXL-50 The proof mass of the ADXL-50 is many times larger than the effective mass of its suspension beams Can ignore the mass of the suspension beams (which greatly simplifies the analysis) Suspension Beam: L = 260 μm, h = 2.3 μm, W = 2 μm Proof Mass Sense Finger Suspension Beam in Tension EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/

10 Lumped Spring-Mass Approximation Mass is dominated by the proof mass 60% of mass from sense fingers Mass = M = 162 ng (nano-grams) Suspension: four tensioned beams Include both bending and stretching terms [A.P. Pisano, BSAC Inertial Sensor Short Courses, ] EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 19 Bending contribution: ADXL-50 Suspension Model Stretching contribution: Total spring constant: add bending to stretching EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/

11 ADXL-50 Resonance Frequency Using a lumped mass-spring approximation: On the ADXL-50 Data Sheet: f o = 24 khz Why the 10% difference? Well, it s approximate plus Above analysis does not include the frequency-pulling effect of the DC bias voltage across the plate sense fingers and stationary sense fingers EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 21 Distributed Mechanical Structures Vibrating structure displacement function: Maximum displacement function (i.e., mode shape function) Seen when velocity y(x,t) = 0 ŷ(x) Procedure for determining resonance frequency: Use the static displacement of the structure as a trial function and find the strain energy W max at the point of maximum displacement (e.g., when t=0, π/ω, ) Determine the maximum kinetic energy when the beam is at zero displacement (e.g., when it experiences its maximum velocity) Equate energies and solve for frequency EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/

12 Displacement: Velocity: Maximum Kinetic Energy y( x, t) = yˆ( x)cos[ ωt] y( x, t) v( x, t) = = ωyˆ( x)sin[ ωt] t At times t = π/(2ω), 3π/(2ω), y( x, t) = 0 Velocity topographical mapping The displacement of the structure is y(x,t) = 0 The velocity is maximum and all of the energy in the structure is kinetic (since W=0): v( x, π ( nω)) = ωyˆ( x) EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 23 Maximum Kinetic Energy (cont) At times t = π/(2ω), 3π/(2ω), y( x, t) = 0 Velocity: Maximum kinetic energy: v( x, π ( nω)) = ωyˆ( x) 1 2 dk = dm [ v( x, t)] 2 dm = ρ( Wh dx) EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/

13 The Raleigh-Ritz Method Equate the maximum potential and maximum kinetic energies: Rearranging yields for resonance frequency: ω = resonance frequency W max = maximum potential energy ρ = density of the structural material W = beam width h = beam thickness ŷ(x) = resonance mode shape EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 25 Example: Folded-Beam Resonator Folded-beam suspension Derive an expression for the resonance frequency of the folded-beam structure at left. Shuttle w/ mass M s Anchor h = thickness Folding truss w/ mass M t \2 EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/

14 Get Kinetic Energies Folded-beam suspension Shuttle w/ mass M s Anchor h = thickness Folding truss w/ mass M t \2 EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/07 27 Get Kinetic Energies (cont) Folded-beam suspension Shuttle w/ mass M s Anchor h = thickness Folding truss w/ mass M t \2 EE C245: Introduction to MEMS Design Lecture 17 C. Nguyen 10/23/