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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: [email protected] Lin: [email protected]
•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.)
EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 6
Internal Bending Moment
Small section of a beam bent in response to a
transverse load R
4
EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 7
Differential Beam Bending Equation
Neutral axis of a bent cantilever beam
EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/07 8
Cantilever Beam w/ a Concentrated Load
F
x L
x
Clamped end condition:At x=0:
w=0dw/dx = 0
Free end condition
h
5
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
EE C245: Introduction to MEMS Design Lecture 4 C. Nguyen 10/16/2007 12
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.]
EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 16
Folded-Flexure Suspensions
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EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 17
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
EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 18
Beam End Conditions
[From Reddy, Finite Element Method]
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EE C245: Introduction to MEMS Design Lecture 7 C. Nguyen 10/10/07 19
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 7 C. Nguyen 10/10/07 20
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 ==