Eigenvalues, Resonance Poles, and Damping in MEMS

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1 Eigenvalues,, and Damping in D. Bindel Computer Science Division Department of EECS University of California, Berkeley Matrix computations seminar, 19 Apr 26

2 Outline

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4 Resonating Radio signal Capacitive drive Mechanical filter Filtered signal Capacitive sense Mechanical high-frequency (high MHz-GHz) filter Your cell phone is mechanical! New filters can be integrated with circuitry = smaller and lower power Can also make frequency references, resonant sensors,...

5 Resonant : Checkerboard D+ D D+ D S S+ S+ S

6 Resonant : Disk resonator SiGe disk resonators built by E. Quévy

7 Resonant : Shear ring hw = 5.e Value = 2.66E+7 Hz

8 Transfer Functions Time domain: Mu + Bu + Ku = bφ(t) y(t) = p T u Laplace domain: s 2 Mû + sbû + K û = b ˆφ(s) ŷ(s) = p T û Transfer function: H(s) = p T (s 2 M + sb + K ) 1 b ŷ(s) = H(s) ˆφ(s)

9 Think Globally, Approximate Locally Want to treat subsystems like simple circuits Pick a simple circuit topology Pick a target resonant frequency ω Approximate H(s) near iω by circuit transfer function

10 Example: Equivalent Circuit for Disk (Clark, Hsu, Abdelmoneum, Nguyen. J 11 (6))

11 Damping and Q Designers want high quality of resonance (Q) Dimensionless damping in a one-dof system d 2 u du + Q 1 dt2 dt + u = F (t) For a resonant mode with frequency ω C: Q := ω 2 Im(ω) = Stored energy Energy loss per radian

12 Damping Mechanisms Possible loss mechanisms: Fluid damping Material losses Thermoelastic damping Anchor loss

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14 ... Where Angels Fear to Tread Wait a moment: When is a domain effectively semi-infinite? What do the model modes mean? How do I estimate the overall accuracy?

15 Simple String One-dimensional wave problem: Solutions have the form s 2 ρu = σ x, x (, 1) σ = Eu x u(, s) = u (s) u(1, s) =. u(x, s) = d sin k(x 1) d = 1/ sin(k) s = iω k = ω ρ/e = ω/c.

16 Dynamic Stiffness Dynamic stiffness at x = is where k = ω/c = ω ρ/e κ(s) := σ(, s) u(, s) = Ek cot(k) at k = nπ, n.

17 Viscoelastic String Now make the constitutive relation hysteretic: s 2 ρu = σ x, x (, 1) σ = E(s)u x u(, s) = u (s) u(1, s) =. where E(s) has poles only on the negative real s axis. Free vibrations solve a nonlinear eigenproblem.

18 Dynamic Stiffness As before, dynamic stiffness is κ(s) := σ(, s) u(, s) = E(s)k cot(k) but now k = ω/c = ω ρ/e(iω).

19 Perturbation for Isolated Singularities Where E(iω) E, E a real constant, can approximate (isolated) poles by perturbation. at where ω ω (1 + 2 tan δ) ω = nπc = nπ E /ρ tan δ = Im(E(iω )) Re(E(iω )).

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21 Long Viscoelastic String Now change length: s 2 ρu = σ x, x (, L) σ = E(s)u x u(, s) = u (s) u(l, s) =. Dynamic stiffness becomes κ L = Ek cot(lk)

22 Dynamic Stiffness as L Consider asymptotic behavior: κ L = Ek cot(lk) = ike eikl + e ikl e ikl e { ikl ike(1 + O(e = 2ikL )) ike( 1 + O(e 2ikL )) Pointwise convergence almost everywhere to κ = { ike, Im(k) > ike, Im(k) < No convergence along Im(k) =.

23 Dynamic Stiffness as L Limiting function is where k + is the root of κ = ik + E k 2 = ω 2 E(iω) ρ chosen so that Im(k + ) >. Have a branch cut along ω 2 E(iω) >.

24 From Long to Infinite As L, eigenvalues cluster along a curve Away from this curve, κ L converges pointwise to κ Curve is a branch cut in definition of κ

25 Large Absorbing 1 L+1 s 2 ρ 1 u(x, s) = σ(x, s), x (, 1) s 2 ρ 2 u(x, s) = σ(x, s), x (1, L + 1) σ(x, s) = E 1 (s)u x (s), x (, 1) σ(x, s) = E 2 (s)u x (s), x (1, L + 1) u(, s) = u (s) u(, L + 1) = And continuity of u and σ across x = 1.

26 Dynamic Stiffness where κ L = E 1 k 1 cos(k 1 ) + i ξ sin(k 1 ) sin(k 1 ) i ξ cos(k 1 ). δ = e 2ik 2L ξ = E 1k 1 E 2 k 2 = ξ = ξ 1 δ 1 + δ. ρ 1 E 1 ρ 2 E 2 If ξ =, recover the clamped case.

27 Asymptotic Behavior Choose k 2 of κ satisfy: such that Im(k 2 ) <. As L, δ, and κ = E 1 k 1 cos(k 1 ) + iξ sin(k 1 ) sin(k 1 ) iξ cos(k 1 ). tan(k 1 ) = iξ Taylor expand to approximate pole location: k 1 nπ + iξ. where ξ is the value of ξ at k 1 = nπ.

28 We chose a principle value for k 2 but the singularity occurs on the other sheet!

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30 Steps Isolated poles near real ω = simple approximation As L, poles cluster As L, convergence to infinite domain a.e. Infinite domain has branch cuts and resonance poles Isolated resonance poles near real ω = simple approximation

31 Infinite via PML 1 Outgoing exp( i x) 1 Incoming exp(i x) Transformed coordinate Re( x)

32 Infinite via PML Outgoing exp( i x) Incoming exp(i x) Transformed coordinate Re( x)

33 Infinite via PML Outgoing exp( i x) Incoming exp(i x) Transformed coordinate Re( x)

34 Infinite via PML Outgoing exp( i x) Incoming exp(i x) Transformed coordinate Re( x)

35 Infinite via PML Outgoing exp( i x) Incoming exp(i x) Transformed coordinate Re( x)

36 Infinite via PML Outgoing exp( i x) Incoming exp(i x) Transformed coordinate Re( x)

37 Take inhomogeneous problem from before and add PML: 1 For real ω, same limiting dynamic stiffness But the branch cut is in a different place! Means resonance poles become true poles L+1

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39 Conclusions Modal analysis gives good approximations near isolated poles Large, damped domains = clustered poles Away from poles, converges to unbounded problem problem has branch cuts, resonance poles PML moves branch cuts, reveals the resonance poles Analysis gives good approximations to original problem near isolated resonance poles