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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
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Stress Gradients in CantileversStress 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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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
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length increases slightly to relieve average stress → induces moment
that bends beam
Stress Gradients in Cantilevers (cont)
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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.]
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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
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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
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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
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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
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Folded-Flexure Suspension Variants
• Below: just a subset of the different versions• All can be analyzed in a similar fashion
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[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 ==
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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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