EE C245 – ME C218Introduction to MEMS Design

Fall 2007Fall 2007

Prof Clark T C NguyenProf. Clark T.-C. Nguyen

Dept of Electrical Engineering & Computer SciencesDept. of Electrical Engineering & Computer SciencesUniversity of California at Berkeley

Berkeley, CA 94720y

L t 15 B C b

EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 1

Lecture 15: Beam Combos

Lecture Outline

• Reading: Senturia, Chpt. 9g p• Lecture Topics:

Bending of beamsCantilever beam under small deflectionsCantilever beam under small deflectionsCombining cantilevers in series and parallelFolded suspensionspDesign implications of residual stress and stress gradients

EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 2

Stress Gradients in CantileversStress Gradients in Cantilevers

EE C245: Introduction to MEMS Design Lecture 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 z-direction

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Stress Gradients in Cantilevers

• Below: surface micromachined cantilever deposited at a high temperature then cooled → assume compressive stress

Average stress

After which,

stress is

Stress gradient Once released, beam length increases slightly

But stress gradient remains

i d t

relieved

EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 5

length increases slightly to relieve average stress → induces moment

that bends beam

Stress Gradients in Cantilevers (cont)

EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 6

Radius of Curvature f/ Stress Gradient

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Measurement of Stress Gradient

• Use cantilever beamsStrain gradient (Γ = slope of stress-thickness curve)

b d fl dcauses beams to deflect up or downAssuming linear strain gradient Γ, z = ΓL2/2

[P. Krulevitch Ph.D.]

EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 9

Tip Bending Distance

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Folded-Flexure SuspensionsFolded Flexure Suspensions

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Folded-Beam Suspension

• Use of folded-beam suspension brings many benefitsStress relief: folding truss is free to move in y-di i b d d dil direction, so beams can expand and contract more readily to relieve stressHigh y-axis to x-axis stiffness ratio Folding TrussHigh y axis to x axis stiffness ratio Folding Truss

y

x

y

zz

EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 12

Comb-Driven Folded Beam Actuator

Beam End Conditions

[From Reddy, Finite

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[ y,Element Method]

Common Loading & 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 Lecture 15 C. Nguyen 10/16/08 14

Series Combinations of Springs

• For springs in series w/ one loadDeflections addS i bi lik “ i i ll l”Spring constants combine like “resistors in parallel”

z x

y

z

y

EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 15

Parallel Combinations of Springs

• For springs in parallel w/ one loadLoad is shared between the two springsS i i h f h i di id l i Spring constant is the sum of the individual spring constants

z x

y

z

y

EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 16

Folded-Flexure Suspension Variants

• Below: just a subset of the different versions• All can be analyzed in a similar fashion

EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 17

[From Michael Judy, Ph.D. Thesis, EECS, UC Berkeley, 1994]

Deflection of Folded Flexures

This equivalent to two cantilevers of

l th L /2length Lc/2

Composite cantilever free ends attach here free ends attach here

Half of F absorbed in other half 4 sets of these pairs each of

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other half (symmetrical)

4 sets of these pairs, each of which gets ¼ of the total force F

Constituent Cantilever Spring Constant

• From our previous analysis:

( )yFyLF ⎟⎞

⎜⎛ 2

2 ( )yLEI

yFLyy

EILFyx c

z

c

cz

cc −=⎟⎟⎠

⎞⎜⎜⎝

⎛−= 3

631

2)( 2

3EIF• From which the spring constant is:

33

)( c

z

c

cc L

EILx

Fk ==

• Inserting Lc = L/2

3324

)2/(3

LEI

LEI

k zzc ==

EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 19

Overall Spring Constant • Four pairs of clamped-guided beams

In each pair, beams bend in series(A t i fl ibl )

Rigid Truss

(Assume trusses are inflexible)• Force is shared by each pair → Fpair = F/4

Leg

L

FFpair

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