Music of the MicrospheresEigenvalue problems from micro-gyro design

David Bindel

Department of Computer ScienceCornell University

Tufts/Schlumberger, 15 Oct 2013

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G. H. Bryan (18641928)

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G. H. Bryan (18641928)

Elected a Fellow of the Royal Society (1895)Best known for Stability in Aviation (1911)Also did important work in thermodynamics and hydrodynamics

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G. H. Bryan (18641928)

Bryan was a friendly, kindly, very eccentric individual...

(Obituary Notices of the Fellows of the Royal Society)

... 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, 18841927)

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Bryans Experiment

On the beats in the vibrations of a revolving cylinder or bellby G. H. Bryan, 1890

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Bryans 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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The Beat Goes On

99.4

99.6

99.8

100

100.2

100.4

100.6

0 0.2 0.4 0.6 0.8 1

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 2 4 6 8 10 12 14

Free vibrations in a rotating frame (simplified):

q + 2Jq + 20q = 0, J [0 11 0

]Eigenvalue problem:

(2I + 2iJ + 20

)q = 0.

Solutions: 0 . = beating !

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A Small Application

Northrup-Grummond HRG(developed c. 1965early 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

=q1(t) +q2(t)

q2

q1

q2

q1

q + 2Jq + 20q = 0, J [0 11 0

]

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A General Picture

=q1(t) +q2(t)

q2

q1

q2

q1

[q1(t)q2(t)

][cos(t) sin(t)sin(t) cos(t)

] [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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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, u) + 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

cm()ucm(r, z), u

s =

m=0

sm()usm(r, z)

where

cm() = diag ( cos(m), sin(m), cos(m))

sm() = diag ( sin(m), cos(m), sin(m)) .

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

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

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Block Diagonal Structure

Finite element system: Muh + Kuh = 0

K =

Kcc0Kss1

Kcc1Kss2

Kcc2. . .

KssMKccM

Mass has same structure.

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Step II: Perfect Geometry, Rotation

Free vibration problem in weak form

w, b(w,a) + a(w,u) = 0.

wherea = u + 2 u + ( x) + x

Discretize by finite elements as before:

Muh + Cuh + Kuh = 0

where C comes from Coriolis term (2b(w, u)).

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Block Structure of Finite Element Matrix

Discretize 2b(w, u):

C =

C00 C01C10 C11 C12

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

Off-diagonal blocks come from cross-axis sensitivity:

= zez + r =

00z

+cos()x sin()ysin()x + cos()y

0

.Neglect cross-axis effects (O(2/20), 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

mq + 2gJq + kq = 0,

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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 + (R) B.

Write weak form in B0 geometry:

b(w,a) =

B0w a J dB0,

a(w,u) =

B0(w) : C : (u) J dB0,

where J = det(I + F) with F = /R.

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Decomposing

Do Fourier decomposition of , too! Consider case where

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

Then we have selection rules

a(w,u) =

{O(k), |m n| = kp0, otherwise

Similar picture for b.

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Decomposing

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

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Block matrix structure

Ex: p = 2

K =

K0 2 3

K1 2

K2 2

K3 2 K4

2 K53 2 K6

. . .

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Impact of Selection Rules

Fast FEA: Can neglect some wave numbers / blocksAlso 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: (r, ) = (cos(2m), 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: (r, ) = (cos(m), 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 Agrees 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 BindelEffects of Imperfections on Solid-Wave Gyroscope DynamicsProceedings of IEEE Sensors 2013, Nov 36.

Thanks to DARPA MRIG + Sunil Bhave and Laura Fegely.

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