below forms an interferometer. Beam deflection x changes the light - - PowerPoint PPT Presentation

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below forms an interferometer
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below forms an interferometer. Beam deflection x changes the light - - PowerPoint PPT Presentation

Jan 30, 2024 438 likes •1.12k views

The gap between the beam and the substrate below forms an interferometer. Beam deflection x changes the light absorbed, leading to a limit cycle oscillation. The gap between the beam and the substrate below forms an interferometer. Beam

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The gap between the beam and the substrate below forms an interferometer. Beam deflection x changes the light absorbed, leading to a limit cycle oscillation.

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The gap between the beam and the substrate below forms an interferometer. Beam deflection x changes the light absorbed, leading to a limit cycle oscillation.

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The gap between the beam and the substrate below forms an interferometer. Beam deflection x changes the light absorbed, leading to a limit cycle oscillation.

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Limit cycle amplitude This system was shown to support a Hopf Bifurcation:

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We are interested in studying systems of coupled MEMS

  • scillators, each of which is similar to the above.
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We are interested in studying systems of coupled MEMS

  • scillators, each of which is similar to the above.

We therefore seek a simpler alternative to the above model because it is so complicated.

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We are interested in studying systems of coupled MEMS

  • scillators, each of which is similar to the above.

We therefore seek a simpler alternative to the above model because it is so complicated. Required features: A second order z Diff.Eqn. and a first order T Diff.Eqn. which supports a limit cycle.

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Here is what we came up with:

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This simplified system supports a limit cycle:

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Comparison of numerical solution versus perturbation solution

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We imagine a system of two such simplified MEMS

  • scillators with spring coupling:
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There’s an in-phase mode, z1=z2=periodic, T1=T2=periodic. Is it stable? For given z0, stability turns out to depend upon the coupling coefficient alpha.

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When viewed in the z1-z2 plane, the in-phase mode appears as a straight line with slope 1:

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We now present the results of numerical integration of the simplified model of two coupled MEMS oscillators: Initial conditions are chosen to be close to the in-phase mode: t=0, z1=0.1, z2=0.11, z1’=z2’=T1=T2=0

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a = 0.10

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a = 0.09

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a = 0.08

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a = 0.07

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a = 0.06

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a = 0.05

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a = 0.05

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a = 0.06

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a = 0.06

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Thus the in-phase mode changes its form somewhere between alpha=0.06 and 0.05. Let’s take a closer look…

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a = 0.057

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a = 0.056

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a = 0.055

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a = 0.054

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a = 0.053

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What happens for even smaller values of alpha?

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a = 0.05

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a = 0.048

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a = 0.046

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a = 0.044

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a = 0.042

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a = 0.040

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a = 0.040

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a = 0.040

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a = 0.039

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a = 0.039

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a = 0.030

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a = 0.020

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So we see that the in-phase mode undergoes a complicated series of changes in form which lead to a type of out-of-phase motion. Now let’s see what happens if initial conditions are chosen to be close to the out-of-phase mode: t=0, z1=0.1, z2= - 0.11, z1’=z2’=T1=T2=0

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a = 0.039

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a = 0.1

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a = 0.8

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a = 0.82

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a = 0.823

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a = 0.823

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a = 0.824

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Stability of the (nonlinear) modes

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Stability of the (nonlinear) modes

In this region both modes are stable

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a = 0.07

Initial conditions: z1(0) = 0.1, z2(0) = 0.00209816, other i.c. are zero

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a = 0.07

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a = 0.07

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All of this has been for the simplified model. But simulations of the original model show even more complicated behavior:

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Locking + entrainment in space of drive amplitude and coupling strength

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Summary

We have presented a simplified model of a MEMS oscillator, and we have investigated the dynamics of a system of two such coupled

  • scillators.
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Summary

We have presented a simplified model of a MEMS oscillator, and we have investigated the dynamics of a system of two such coupled

  • scillators.

This work was supported by NSF grant ACI-1548562 NSF grant CMMI 1634664

Acknowledgement

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FIN