EE C245 - ME C218 Introduction to MEMS Design Fall Today s Lecture

Transcription

1 EE C45 - ME C8 Introduction to MEMS Design Fall 3 Roger Howe and Thara Srinivasan Lecture 9 Energy Methods II Today s Lecture Mechanical structures under driven harmonic motion develop analytical techniques for estimating the resonant frequency Lumped mass-spring case: ADXL-5 example Rayleigh-Ritz method and examples: lateral resonator, double-ended tuning fork Viscous damping Damped nd order mechanical systems: transfer functions Reading: Senturia, S. D., Microsystem Design, Kluwer Academic Publishers,, Chapter 9, pp

2 Estimating Resonant Frequencies Simple harmonic motion k M x(t) = x o cos(ωt) Potential energy: W (t) = Kinetic energy: K (t) = 3 Energy Conservation Ignoring dissipation W max = K max Resonant frequency: Applications: lumped masses that dominate the suspension s mass 4

3 Example: ADXL-5 Suspension beam: L = 6 µm, h =.3 µm, W = µm Average residual stress through thickness: σ r = 5 MPa 5 Lumped Spring-Mass Approximation Mass = M = 6 nano-grams, 6% is from the capacitive sense fingers Suspension: four tensioned beams include both bending and stretching terms F/4 Bending compliance k b - F/4 Stretching compliance k st - A.P. Pisano, BSAC Inertial Sensor Short Courses,

4 ADXL-5 Suspension Model Bending contribution: Stretching contribution: S θ S Total spring constant: A.P. Pisano, BSAC Inertial Sensor Short Courses, ADXL-5 Resonant Frequency Lumped mass-spring approximation: Data sheet: f = 4 khz A.P. Pisano, BSAC Inertial Sensor Short Courses,

5 Distributed Mechanical Structures Vibrating structure s displacement function can be separated into two parts: We can use the static displacement of the structure as a trial function and find the strain energy W max at the point of maximum displacement (t =, π/ω, ) y(x) 9 Maximum Kinetic Energy At times t = π/(ω), 3π/(ω),, then the displacement of the structure is y(x) =. The velocity of the structure is maximum and all its energy is kinetic (since W = ) y(x,t ) = v(x,t ) 5

6 Finding the Maximum Kinetic Energy A differential length dx along the beam has a kinetic energy dk due to its transverse motion: v(x,t ) = - ωy(x) W h dx Total maximum kinetic energy: The Raleigh-Ritz Method The maximum potential and maximum kinetic energies must be equal The resonant frequency of the beam is therefore: 6

7 Example: Folded-Flexure Resonator h W h T Lumped masses (shuttle, truss) need to be included in the kinetic energy expression 3 Kinetic Energy Shuttle mass = M s, truss mass = M t L Kmax = M svs + Mtvt + ρwhω yˆ ( x) dx Use static deflection as estimate of mode shape x yˆ( x) y( x) = Yo 3 L x L 3 Magnitude of shuttle velocity is v s =ωy o Magnitude of truss velocity is v t = ωy o / 4 7

8 Resonant Frequency Find maximum potential energy from spring constant and shuttle deflection: W = max k y Y o Rayleigh-Ritz equation: k 4EI L z y = = 3 EWh 3 L 3 ω = M s + k y ( M 4) + (M /35) t / b both trusses all 8 beams 5 Double-Ended Tuning Forks Trey Roessig, Ph.D., ME Dept., UC Berkeley,

9 Mode Shapes for Clamped-Clamped Beams φ (ε =.5) = φ,max = φ 3 (ε =.5) = φ ( ε ) = κ[sinh( βε sin( βε) + α(cosh( βε) cos( βε))] Mode α β κ [ε = x/l] First Second Third Applying Rayleigh-Ritz to DETFs Case : σ r = Case : σ r = 3 MPa model comb drive as a point mass with m = pgm Dimensions: L = µm, h = W = µm, E = 5 GPa, ρ =.3 gmcm -3 Use first mode shape for both cases: analyze a single tine Trey Roessig, Ph.D., ME Dept., UC Berkeley,

10 DETF Results Bending: d φ dε = 98.6 dε ω dφ Axial load: dε = 4. dε 85 Kinetic energy (distributed along beam): φ ( ε ) dε =. 397 EI L = z 3 d φ dε σ rwh dε + L ρwhl φ dε + mφ dφ dε (.5) dε k = M eff eff Case : 4 khz Case : 339 khz Trey Roessig, Ph.D., ME Dept., UC Berkeley, Resonant Sensors The double-ended tuning fork is an excellent sensor for axial stress, since it converts it into a shift in frequency (or in oscillation period) these are easily and accurately measured. Therefore, we can use the DETF as a building block for a variety of resonant sensors, such as accelerometers (T. A. Roessig, Ph.D. ME, 998) and gyroscopes (A. A. Seshia, Ph.D. EECS, ) Rayleigh-Ritz Method yields analytical expressions for frequency shift due to axial stress (residual or applied): 4.85 dω dσ / k dk eff eff eff r d = dσ r M eff k = M eff dσ r = ω Wh L dφ dε dε design insight!

11 Finite Element Analysis Mark Lemkin, Ph.D., ME Dept., UC Berkeley, 997 Damping Many sources: ignore internal damping and acoustic radiation from the anchors focus on drag from the surrounding gas Gap dimension can be on the order of mean-free path continuum model doesn t apply David Horsley, Ph.D., ME Dept., UC Berkeley, 998

12 Couette Damping Drag force = F d = -bv y Damping coefficient: b b = µa / y o Effective viscosity at reduced pressure (p << 5 Torr): µ= (µ p ) py o = (3.7 x - ) py o units fixed units of µ p are kg-s - -m - Torr - for p in Torr William Clark, Ph.D., EECS Dept., UC Berkeley, Squeeze-Film Damping Plates slide in y direction b 7µAz o / y o 3 Note that it s just µ, not µ p 4

13 Mass-Spring-Damper System b F(t) = Fcos(ωt) k M x(t) = Xcos(ωt) Sinusoidal, steady-state response: X(t) = X e jωt 5 Second-Order Resonant Transfer Function Solve for the ratio of the phasor displacement X to the phasor driving force F Above analysis is for the light damping case (Q >> ) 6 3

14 Limiting Cases Low frequency: ω << ω High frequency: ω >> ω Resonant frequency: ω = ω Peak width (-3 db): ω = ω / Q 7 Magnitude Bode Plot 6 H db 4 ω - ω ω ω + ω ω - k - = m/n Q = 8 4

15 Phase Bode Plot Phase(H) -45 ω - ω ω ω + ω ω