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EE C245 – ME C218Introduction to MEMS Design
F ll 200Fall 2007
Prof. Clark T.-C. Nguyen
Dept. of Electrical Engineering & Computer SciencesUniversity of California at Berkeley
EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 1
University of California at BerkeleyBerkeley, CA 94720
Lecture 15: Beam Combos
Lecture Outline
• Reading: Senturia, Chpt. 9• Lecture Topics:
Bending of beamsBending of beamsCantilever beam under small deflectionsCombining cantilevers in series and parallelFolded suspensionsDesign implications of residual stress and stress gradients
EE C245: Introduction to MEMS Design Lecture 15 C. Nguyen 10/16/08 2
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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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Stress Gradients in Cantilevers
• Below: surface micromachined cantilever deposited at a high temperature then cooled → assume compressive stress
Average stress
After
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Stress gradient Once released, beam length increases slightly to relieve average stress
But stress gradient remains → induces moment that bends beam
which, stress is relieved
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) causes beams to deflect up or downAssuming linear strain gradient Γ, z = ΓL2/2g g ,
[P. Krulevitch Ph.D.]
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Tip Bending Distance
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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 Folding Truss
x
y
z
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Comb-Driven Folded Beam Actuator
Beam End Conditions
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[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”
x
y
z
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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
x
y
z
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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]
9
Deflection of Folded Flexures
This equivalent to two cantilevers of
length Lc/2
Composite cantilever Composite cantilever free ends attach here
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Half of F absorbed in 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:
( )yLEI
yFLyy
EILFyx c
z
c
cz
cc −=⎟⎟⎠
⎞⎜⎜⎝
⎛−= 3
631
2)(
22
zcz ⎠⎝
33
)( c
z
c
cc L
EILx
Fk ==
• From which the spring constant is:
• Inserting L = L/2
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Inserting Lc = L/2
3324
)2/(3
LEI
LEI
k zzc ==
10
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
Rigid Truss
Force is shared by each pair → Fpair F/4
Leg
L
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Fpair
Folded-Beam Stiffness Ratios
• In the x-direction:
• In the z direction:
Folded-beam suspension
324
LEI
k zx =
In the z-direction:Same flexure and boundary conditions
• In the y-direction:Shuttle
324
LEIk x
z =
EWh8
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• Thus: Anchor
Folding truss
LEWhk y
8=
2
4 ⎟⎠⎞
⎜⎝⎛=WL
kk
x
yMuch
stiffer in y-direction!
[See Senturia, §9.2]
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Folded-Beam Suspensions Permeate MEMS
Gyroscope [Draper Labs.]Accelerometer [ADXL-05, Analog Devices]
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Micromechanical Filter [K. Wang, Univ. of Michigan]
Folded-Beam Suspensions Permeate MEMS
• Below: Micro-Oven Controlled Folded-Beam Resonator
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Stressed Folded-Flexures
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Clamped-Guided Beam Under Axial Load
• Important case for MEMS suspensions, since the thin films comprising them are often under residual stress
• Consider small deflection case: y(x) « Lxz
Governing differential equation: (Euler Beam Equation)
L
W
x
y
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Governing differential equation: (Euler Beam Equation)
Unit impulse @ x=LAxial Load
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The Euler Beam Equation
• Axial stresses produce no net horizontal 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 axial stress derived force
RAxial Stress
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on the beam to counteract the axial stress-derived forceFor ease of analysis, assume the beam is bent to angle π
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
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Clamped-Guided Beam Under Axial Load
• Important case for MEMS suspensions, since the thin films comprising them are often under residual stress
• Consider small deflection case: y(x) « Lxz
Governing differential equation: (Euler Beam Equation)
L
W
x
y
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Governing differential equation: (Euler Beam Equation)
Unit impulse @ x=LAxial Load
Solving the ODE
• Can solve the ODE using standard methodsSenturia, pp. 232-235: 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. p gTimoshenko, Strength of Materials II: Advanced Theory and Problems, McGraw-Hill, New York, 3rd Ed., 1955
• Result from Timoshenko:
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Design Implications
• Straight flexuresLarge tensile S means flexure behaves like a tensioned wire (for which k-1 = L/S)Large compressive S can lead to buckling (k-1 → ∞)g p g ( )
• Folded flexuresResidual stress only partially releasedLength from truss to shuttle’s centerline differs
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centerline differs by Ls for inner and outer legs
Effect on Spring Constant
• Residual compression on outer legs with same magnitude of tension on inner legs:
⎟⎠⎞
⎜⎝⎛±=
LLs
rb εεBeam Strain: ; Stress Force: WhLLES s
r ⎟⎠⎞
⎜⎝⎛±= ε
• Spring constant becomes:⎠⎝ Lrb Lr
⎠⎝
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• Remedies:Reduce the shoulder width Ls to minimize stress in legsCompliance in the truss lowers the axial compression and tension and reduces its effect on the spring constant