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 ecture 15: Beam Combos EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/08 1 ecture Outline Reading: Senturia, Chpt. 9 ecture Topics: Bending of beams Cantilever beam under small deflections Combining cantilevers in series and parallel Folded suspensions Design implications of residual stress and stress gradients EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/08 2 1

2 Stress Gradients in Cantilevers EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/08 3 Vertical Stress Gradients Variation of residual stress in the direction of film growth Can warp released structures in -direction EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/08 4 2

3 Stress Gradients in Cantilevers Below: surface micromachined cantilever deposited at a high temperature then cooled assume compressive stress Average stress Stress gradient Once released, beam length increases slightly to relieve average stress After which, stress is relieved But stress gradient remains induces moment that bends beam EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/08 5 Stress Gradients in Cantilevers (cont) EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/08 6 3

4 Radius of Curvature f/ Stress Gradient EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/08 7 Measurement of Stress Gradient Use cantilever beams Strain gradient (Γ = slope of stress-thickness curve) causes beams to deflect up or down Assuming linear strain gradient Γ,, = Γ 2 /2 [P. Krulevitch Ph.D.] EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/08 8 4

5 Tip Bending Distance EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/08 9 Folded-Fleure Suspensions EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/

6 Folded-Beam Suspension Use of folded-beam suspension brings many benefits Stress relief: folding truss is free to move in y- direction, so beams can epand and contract more readily to relieve stress High y-ais to -ais stiffness ratio Folding Truss y Comb-Driven Folded Beam Actuator EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/08 11 Beam End Conditions [From Reddy, Finite Element Method] EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/

7 Common oading & Boundary Conditions Displacement equations derived for various beams with concentrated load F or distributed load f Gary Fedder Ph.D. Thesis, EECS, UC Berkeley, 1994 EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/08 13 Series Combinations of Springs For springs in series w/ one load Deflections add Spring constants combine like resistors in parallel y EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/

8 Parallel Combinations of Springs For springs in parallel w/ one load oad is shared between the two springs Spring constant is the sum of the individual spring constants y EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/08 15 Folded-Fleure Suspension Variants Below: just a subset of the different versions All can be analyed in a similar fashion [From Michael Judy, Ph.D. Thesis, EECS, UC Berkeley, 1994] EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/

9 Deflection of Folded Fleures This equivalent to two cantilevers of length c /2 Composite cantilever free ends attach here Half of F absorbed in other half (symmetrical) 4 sets of these pairs, each of which gets ¼ of the total force F EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/08 17 Constituent Cantilever Spring Constant From our previous analysis: 2 Fc c y Fc y y = 2 ( ) y = EI c 6EI F k c = = ( y) From which the spring constant is: Inserting c = /2 c 3EI c 3 ( c ) c k 3EI c = = 3 ( / 2) 24EI 3 EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/

10 Overall Spring Constant Rigid Truss Four pairs of clamped-guided beams In each pair, beams bend in series (Assume trusses are infleible) Force is shared by each pair F pair = F/4 eg F pair EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/08 19 Folded-Beam Stiffness Ratios Folded-beam suspension Anchor Shuttle Folding truss In the -direction: 24EI 3 In the -direction: Same fleure and boundary conditions k k = = In the y-direction: [See Senturia, 9.2] Thus: k k y = 4 W 24EI 3 8EWh k y = 2 Much stiffer in y-direction! EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/

11 Folded-Beam Suspensions Permeate MEMS Accelerometer [ADX-05, Analog Devices] Gyroscope [Draper abs.] Micromechanical Filter [K. Wang, Univ. of Michigan] EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/08 21 Folded-Beam Suspensions Permeate MEMS Below: Micro-Oven Controlled Folded-Beam Resonator EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/

12 Stressed Folded-Fleures EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/08 23 Clamped-Guided Beam Under Aial oad Important case for MEMS suspensions, since the thin films comprising them are often under residual stress Consider small deflection case: y() «y W Governing differential equation: (Euler Beam Equation) Aial oad Unit = EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/

13 The Euler Beam Equation Aial Stress R Aial stresses produce no net horiontal force; but as soon as the beam is bent, there is a net downward force For equilibrium, must postulate some kind of upward load on the beam to counteract the aial stress-derived force For ease of analysis, assume the beam is bent to angle π EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/08 25 The Euler Beam Equation Note: Use of the full bend angle of π to establish conditions for load balance; but this returns us to case of small displacements and small angles EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/

14 Clamped-Guided Beam Under Aial oad Important case for MEMS suspensions, since the thin films comprising them are often under residual stress Consider small deflection case: y() «y W Governing differential equation: (Euler Beam Equation) Aial oad Unit = EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/08 27 Solving the ODE Can solve the ODE using standard methods Senturia, pp : solves ODE for case of point load on a clamped-clamped beam (which defines B.C. s) For solution to the clamped-guided case: see S. Timoshenko, Strength of Materials II: Advanced Theory and Problems, McGraw-Hill, New York, 3 rd Ed., 1955 Result from Timoshenko: EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/

15 Design Implications Straight fleures arge tensile S means fleure behaves like a tensioned wire (for which k -1 = /S) arge compressive S can lead to buckling (k -1 ) ) Folded fleures Residual stress only partially released ength from truss to shuttle s centerline differs by s for inner and outer legs EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/08 29 Effect on Spring Constant Residual compression on outer legs with same magnitude of tension on inner legs: s Beam Strain: ; Stress Force: s S = ± Eε r Wh ε = ± b ε r Spring constant becomes: Remedies: Reduce the shoulder width s to minimie stress in legs Compliance in the truss lowers the aial compression and tension and reduces its effect on the spring constant EE C245: Introduction to MEMS Design ecture 15 C. Nguyen 10/16/