EE C245 ME C218 Introduction to MEMS Design

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 19: Resonance Frequency II EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 1

2 Lecture Outline Reading: Senturia, Chpt. 10 Lecture Topics: Energy Methods Virtual Work Energy Formulations Tapered p Beam Example Estimating Resonance Frequency EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 2

3 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 19 C. Nguyen 11/4/08 3

4 Example: Folded-Beam Resonator Folded-beam suspension Derive an expression for the resonance frequency of the folded-beam d structure at left. Shuttle w/ mass M s Anchor h = thickness Folding truss w/ mass M t \2 t EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 4

5 Get Kinetic Energies Folded-beam suspension Shuttle w/ mass M s Anchor h = thickness Folding truss w/ mass M t \2 t EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 5

6 Folded-Beam Suspension Folding Truss y z x Comb-Driven Folded Beam Actuator EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 6

7 Get Kinetic Energies (cont) Folded-beam suspension Shuttle w/ mass M s Anchor h = thickness Folding truss w/ mass M t \2 t EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 7

8 Get Kinetic Energies (cont) Folded-beam suspension Shuttle w/ mass M s Anchor h = thickness Folding truss w/ mass M t \2 t EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 8

9 Get Potential Energy & Frequency Folded-beam suspension Shuttle w/ mass M s Anchor h = thickness Folding truss w/ mass M t \2 t EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 9

10 Brute Force Methods for Resonance Frequency Determination ti EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 10

11 Basic Concept: Scaling Guitar Strings Guitar String μmechanical M h i l Resonator Guitar Vib. Am mplitud e Freq. Low Q High Q 110 Hz Freq. Vibrating A String (110 Hz) Stiffness Freq. Equation: 1 kr f o = 2π m r Mass [Bannon 1996] Performance: L r =40.8μm f o =8.5MHz m r ~ kg Q vac =8,000 W r =8μm, h r =2μm Q air ~50 d=1000å, V P =5V Press.=70mTorr EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 11

12 Fixed-Fixed Beam Resonator Anchor Anchor Losses Gap Electrode Anchor Q = 300 at 70MHz Free-Free Beam Resonator Anchor Supporting Beams Elastic Wave Radiation Problem: direct anchoring to the substrate anchor radiation into the substrate lower Q Solution: support at motionless nodal points isolate resonator from anchors less energy loss higher Q L r Anchor Free-Free Beam Q = 15,000 at 92MHz EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 12

13 92 MHz Free-Free Beam μresonator Free-free beam μmechanical h l resonator with non-intrusive supports reduce anchor dissipation higher Q EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 13

14 Higher Order Modes for Higher Freq. 2 nd Mode Free-Free Beam 3 rd Mode Free Free Beam Distinct Mode Shapes Electrodes Ancho r μm Ancho r h = 2.1 μm Support Beam B] Transm mission [d -57 = ]Lr Q = 11, Frequency [MHz] EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 14 Phas se [degree

15 Flexural-Mode Beam Wave Equation y z u Transverse Displacement W= width u = ma L h x F F + F x dx Derive the wave equation for transverse vibration: EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 15

16 Example: Free-Free Beam z W h Determine the resonance frequency of the beam Specify the lumped parameter mechanical equivalent circuit Transform to a lumped parameter electrical equivalent circuit Start with the flexural-mode beam equation: 2 t u 2 = EI ρ A 4 x u 4 EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 16

17 Free-Free Beam Frequency Substitute u = u 1 e jωt into the wave equation: (1) This is a 4 th order differential equation with solution: (2) Boundary Conditions: EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 17

18 Free-Free Beam Frequency (cont) Applying B.C. s, get A=C and B=D, and (3) Setting the determinant = 0 yields Which has roots at Substituting (2) into (1) finally yields: Free-Free Beam Frequency Equation EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 18

19 Higher Order Free-Free Beam Modes More than 10x increase Fundamental Mode (n=1) 1 st Harmonic (n=2) 2 nd Harmonic (n=3) EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 19

20 Mode Shape Expression The mode shape expression can be obtained by using the fact that A=C and B=D into (2), yielding Get the amplitude ratio by expanding (3) [the matrix] and solving, which yields Then just substitute the roots for each mode to get the expression for mode shape Fundamental Mode (n=1) [Substitute ] EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 20

21 Lumped Parameter Mechanical Equivalent Circuit it EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 21

22 Equivalent Dynamic Mass Once the mode shape is known, the lumped parameter equivalent circuit can then be specified Determine the equivalent mass at a specific location x using knowledge of kinetic energy and velocity z Location x W h Maximum Kinetic Energy Density Equivalent Mass = Maximum location x Maximum Velocity Function EE C245: Introduction to MEMS Design Lecture 19 C. Nguyen 11/4/08 22