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EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 1

EE C245 – ME C218Introduction to MEMS Design

Fall 2007

Prof. Clark T.-C. Nguyen

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

Berkeley, CA 94720

Lecture 15: Beam Combos

EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 2

Announcements

• Turn part (a) of your HW#4 in as an emailed gds file to the TA of your choice

Li-Wen Hung: lwhung@eecs.berkeley.eduYang Lin: linyang@eecs.berkeley.edu

•Midterm Exam will be Thursday, Nov. 1

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EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 3

Lecture Outline

• Reading: Senturia Chpts. 9, 10• Lecture Topics:

Beam BendingStress Gradients in CantileversFolded-Flexure SuspensionsStressed Folded-FlexuresEnergy Methods

Virtual WorkEnergy Formulations

EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 4

Beam Segment in Pure Bending

Small section of a beam bent in response to a tranverse load R

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EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 5

Beam Segment in Pure Bending (cont.)

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Internal Bending Moment

Small section of a beam bent in response to a

transverse load R

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Differential Beam Bending Equation

Neutral axis of a bent cantilever beam

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Cantilever Beam w/ a Concentrated Load

F

x L

x

Clamped end condition:At x=0:

w=0dw/dx = 0

Free end condition

h

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EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 9

Cantilever Beam w/ a Concentrated Load

F

x L

x

Clamped end condition:At x=0:

w=0dw/dx = 0

Free end condition

h

EE C245: Introduction to MEMS Design Lecture 2 C. Nguyen 8/30/07 10

Maximum Stress in a Bent Cantilever

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EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 11

Stress Gradients in Cantilevers

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Vertical Stress Gradients

• Variation of residual stress in the direction of film growth• Can warp released structures in z-direction

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EE C245: Introduction to MEMS Design Lecture 2 C. Nguyen 8/30/07 13

Stress Gradients in Cantilevers

Average stress

Stress gradient

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

Once released, beam length increases slightly to relieve average stress

But stress gradient remains → induces moment that bends beam

After which, stress is relieved

EE C245: Introduction to MEMS Design Lecture 2 C. Nguyen 8/30/07 14

Stress Gradients in Cantilevers (cont)

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EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 15

Measurement of Stress Gradient

• Use cantilever beamsStrain gradient (Γ = slope of stress-thickness curve) causes beams to deflect up or downAssuming linear strain gradient Γ, z = ΓL2/2

[P. Krulevitch Ph.D.]

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Folded-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-direction, so beams can expand and contract more readily to relieve stressHigh y-axis to x-axis stiffness ratio

Comb-Driven Folded Beam Actuator

Folding Truss

x

y

z

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Beam End Conditions

[From Reddy, Finite Element Method]

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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

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Series Combinations of Springs

• For springs in series w/ one loadDeflections addSpring constants combine like “resistors in parallel”

Compliances effectively add:

1/k = 1/kc + 1/kc

Y(L) = F/k = 2 y(Lc) = 2 (F/kc) = F(1/kc + 1/kc)

k = kc||kc→

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EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 21

Parallel Combinations of Springs

• For springs in parallel w/ one loadLoad is shared between the two springsSpring constant is the sum of the individual spring constants

Y(L) = F/k = Fa/ka = Fb/kb = (F/2) (1/ka)

k = 2 ka

EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 22

Folded-Flexure Suspension Variants

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

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

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EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 23

Deflection of Folded Flexures

Half of F absorbed in other half

(symmetrical)

This equivalent to two cantilevers of

length L/2

Composite cantilever free ends attach here

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

EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 24

Constituent Cantilever Spring Constant

• From our previous analysis:

( )yLEI

yFLyy

EILFyx c

z

c

cz

cc −=⎟⎟⎠

⎞⎜⎜⎝

⎛−= 3

631

2)(

22

33

)( c

z

c

cc L

EILx

Fk ==

• From which the spring constant is:

• Inserting Lc = L/2

3324

)2/(3

LEI

LEI

k zzc ==

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EE C245: Introduction to MEMS Design Lecture 2 C. Nguyen 8/30/07 25

Overall Spring Constant

• Four pairs of clamped-guided beamsIn each pair, beams bend in series(Assume trusses are inflexible)

• Force is shared by each pair → Fpair = F/4

Fpair

Rigid Truss

Leg

L