Music of the MicrospheresEigenvalue problems from micro-gyro design

David Bindel

Department of Computer ScienceCornell University

Oxford NA seminar, 29 Apr 2014

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The Computational Science & Engineering Picture

Application

Analysis Computation

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A Favorite Application: MEMS

I’ve worked on this for a while:SUGAR (early 2000s) – SPICE for MEMSHiQLab (2006) – high-Q mechanical resonator device modelingAxFEM (2012) – solid-wave gyro device modeling

Goal today: an illustrative snapshot.

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MEMS Basics

Micro-Electro-Mechanical SystemsChemical, fluid, thermal, optical (MECFTOMS?)

Applications:Sensors (inertial, chemical, pressure)Ink jet printers, biolab chipsRadio devices: cell phones, inventory tags, pico radio

Use integrated circuit (IC) fabrication technologyTiny, but still classical physics

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Where are MEMS used?

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My favorite applications

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G. H. Bryan (1864–1928)

Fellow of the Royal Society (1895)Stability in Aviation (1911)Thermodynamics, hydrodynamics

Bryan was a friendly, kindly, veryeccentric individual...(Obituary Notices of the FRS)

... if he sometimes seemed a colossal buffoon, he himself didnot help matters by proclaiming that he did his best workunder the influence of alcohol.(Williams, J.G., The University College of North Wales, 1884–1927)

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Bryan’s Experiment

“On the beats in the vibrations of a revolving cylinder or bell”by G. H. Bryan, 1890

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Bryan’s Experiment Today

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The Beat Goes On

0.00 6.48 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 5.30 5.40 5.50 5.60 5.70 5.80 5.90 6.00 6.10 6.20 6.30 6.40

Audio Track

0.00 8.36 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 5.30 5.40 5.50 5.60 5.70 5.80 5.90 6.00 6.10 6.20 6.30 6.40 6.50 6.60 6.70 6.80 6.90 7.00 7.10 7.20 7.30 7.40 7.50 7.60 7.70 7.80 7.90 8.00 8.10 8.20

Audio Track

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A Small Application

Northrup-Grumman HRG(developed c. 1965–early 1990s)

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A Smaller Application (Cornell)

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A Smaller Application (UMich, GA Tech, Irvine)

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A Smaller Application!

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Micro-HRG / GOBLiT / OMG

Goal: Cheap, small (1mm) HRGCollaborator roles:

Basic designFabricationMeasurement

Our part:Detailed physicsFast softwareSensitivityDesign optimization

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Foucault in Solid State

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Rate Integrating Mode

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Rate Integrating Mode

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A General Picture: Stationary Frame

=q1(t) +q2(t)

q2

q1

Consider free vibration consisting of two modes of oscillation:

q̈ + ω20q = 0, q(t) ∈ R2.

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A General Picture: Rotating Frame

=q1(t) +q2(t)

q2

q1

q2

q1

Rate of rotation is Ω� ω0:

q̈ + 2βΩJq̇ + ω20q = 0, J ≡[0 −11 0

]

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A General Picture: Rotating Frame

=q1(t) +q2(t)

q2

q1

q2

q1

[q1(t)q2(t)

]≈[cos(−βΩt) − sin(−βΩt)sin(−βΩt) cos(−βΩt)

] [q01(t)q02(t)

].

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An Uncritical FEA Approach

Why not do the obvious?Build 3D model with commercial FERun modal analysis

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The Perturbation Picture

Perturbations split degenerate modes:Coriolis forces (good)Imperfect fab (bad, but physical)Discretization error (non-physical)

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The Perturbation Picture

Perturbations split some degeneracies:Zeeman effect (magnetic)Stark effect (electric)

This is far from unique to MEMS!

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Three Step Program

1 Perfect geometry, no rotation2 Perfect geometry, rotation3 Imperfect geometry

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Step I: Perfect Geometry, No Rotation

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Step I: Perfect Geometry, No Rotation

Free vibration problem in weak form

∀w, b(w, ü) + a(w,u) = 0.

where

b(w,a) =

∫B0ρw · a dB0,

a(w,u) =

∫B0ε(w) : C : ε(u) dB0,

Or think:Mü + Ku = 0

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Step I: Perfect Geometry, No Rotation

Free vibration problem in weak form

∀w, b(w, ü) + a(w,u) = 0.

Symmetry: Q any rotation or reflection

b(Qw, Qu) = b(w,u)

a(Qw, Qu) = a(w,u)

Decompose by invariant subspaces of Q =⇒ Fourier analysis

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Fourier Expansion and Axisymmetric Shapes

Decompose into symmetric uc and antisymmetric us in y:

uc =

∞∑m=0

Φcm(θ)ucm(r, z), u

s =

∞∑m=0

Φsm(θ)usm(r, z)

where

Φcm(θ) = diag ( cos(mθ), sin(mθ), cos(mθ))

Φsm(θ) = diag (− sin(mθ), cos(mθ),− sin(mθ)) .

Modes involve only one azimuthal number m; degenerate for m > 1.

Preserve structure in FE: shape functions Nj(r, z)Φc,sm (θ)

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Block Diagonal Structure

Finite element system: Müh + Kuh = 0

K =

Kcc0Kss1

Kcc1Kss2

Kcc2. . .

KssMKccM

Mass has same structure.

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Step II: Perfect Geometry, Rotation

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Step II: Perfect Geometry, Rotation

Free vibration problem in weak form

∀w, b(w,a) + a(w,u) = 0.

wherea = ü + 2Ω× u̇ + Ω× (Ω× x) + Ω̇× x

Assumption: Ω2/ω2 and Ω̇/ω2 are negligible.

Discretize by finite elements as before:

Müh + Cu̇h + Küh = 0

where C comes from Coriolis term (2b(w,Ω× u̇)).

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Block Structure of Finite Element Matrix

Discretize 2b(w,Ω× u̇):

C =

C00 C01C10 C11 C12

C21 C22 C23. . . . . . . . .

Off-diagonal blocks come from cross-axis sensitivity:

Ω = Ωzez + Ωrθ =

00Ωz

+cos(θ)Ωx − sin(θ)Ωysin(θ)Ωx + cos(θ)Ωy

0

.Neglect cross-axis effects (O(Ω2/ω20), like centrifugal effect).

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Analysis in Ideal Case

Only need to mesh a 2D cross-section!Compute an operating mode uc for the non-rotating geometry.Compute associated modal mass and stiffness m and k.Compute g = b(uc, ez × us).Model: motion is approximately q1uc + q2us, and

mq̈ + 2gΩJq̇ + kq = 0,

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Step II: Imperfect Geometry

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Step III: Imperfect Geometry

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What Imperfections?

Let me count the ways...Over/under etchMask misalignmentThickness variationsAnisotropy of etching single-crystal Si

These are not arbitrary!

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Representing the Perturbation

Map axisymmetric B0 → real B:

R ∈ B0 7→ r = R + �ψ(R) ∈ B.

Write weak form in B0 geometry:

b(w,a) =

∫B0ρw · a J dB0,

a(w,u) =

∫B0ε(w) : C : ε(u) J dB0,

where J = det(I + �F) with F = ∂ψ/∂R.

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Decomposing ψ

Do Fourier decomposition of ψ, too! Consider case where

m = only azimuthal number of wn = only azimuthal number of up = only azimuthal number of ψ

Then we have selection rules

a(w,u) =

{O(�k), |m± n| = kp0, otherwise

Similar picture for b.

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Decomposing ψ

Over/under etch (p = 0)Mask misalignment (p = 1)Thickness variations (p = 1)Anisotropy of etching single-crystal Si (p = 3 or p = 4)

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Block matrix structure

Ex: p = 2

K =

K0 � �2 �3

K1 � �2

� K2 � �2

� K3 ��2 � K4 �

�2 � K5�3 �2 � K6

. . .

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Impact of Selection Rules

Fast FEA: Can neglect some wave numbers / blocksAll assuming no accidental (near) degeneraciesFirst order: Only need diagonal blocksSecond order: Diagonal blocks plus “directly coupled”

Also qualitative information

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Qualitative Information

Operating wave number m, perturbation number p:

p = 2m frequencies split by O(�)kp = 2m frequencies split at most O(�2)p 6 |2m frequencies change at O(�2), no splitp = 1

p = 2m± 1 O(�) cross-axis coupling.

Note:m = 2 affected at first order by p = 0 and p = 4(and O(�2) split from p = 1 and p = 2).m = 3 affected at first order by p = 0 and p = 6(and O(�2) split from p = 1 and p = 3).

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Mode Split for Rings: ψ(r, θ) = (cos(2mθ), 0).

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22

−2

0

2

Perturbation amplitude / thickness

Freq

uenc

ysh

ift(%

)

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Mode Split for Rings: ψ(r, θ) = (cos(mθ), 0).

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22

−1

−0.5

0

Perturbation amplitude / thickness

Freq

uenc

ysh

ift(%

)

m = 2a m = 2bm = 3a m = 3bm = 4a m = 4b

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Analyzing Imperfect Rings

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Analyzing Imperfect Rings

q2

q1

q2

q1

q2

q1

q2

q1

q2

q1

q2

q1

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Beyond Rings: AxFEM

Mapped finite / spectral element formulationLow-order polynomials through thicknessHigh-order polynomials along lengthTrig polynomials in θAgrees with results reported in literatureComputes sensitivity to geometry, material parameters, etc.

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Further Steps

Lots of possible directions:Symmetry breaking through damping?Integration with fabrication simulation?Joint optimization of geometry and fabrication?

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Thank You

Yilmaz and Bindel“Effects of Imperfections on Solid-Wave Gyroscope Dynamics”Proceedings of IEEE Sensors 2013, Nov 3–6.

Thanks to DARPA MRIG + Sunil Bhave and Laura Fegely.

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Open problem: Accurate etch simulation

Simple isotropic etch modeling fails – 1mm is huge!Working on better simulator (reaction-diffusion).For now, take idealized geometries on faith...

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Open problem: Appropriate damping models

Bulk waves in Si about 8500 m/s.Typical wafer: 150 mm (6 in) is 675 µmGHz resonator: wavelength of tens of micronskHz resonator: wavelength of centimetersCompletely different anchor models should apply!

Surface losses also matter...

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Appendix