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

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 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 deflection x changes the light absorbed, leading to a limit cycle oscillation.

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

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

We are interested in studying systems of coupled MEMS oscillators, each of which is similar to the above. We therefore seek a simpler alternative to the above model because it is so complicated.

We are interested in studying systems of coupled MEMS oscillators, 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.

Here is what we came up with:

This simplified system supports a limit cycle:

Comparison of numerical solution versus perturbation solution

We imagine a system of two such simplified MEMS oscillators with spring coupling:

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.

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

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

a = 0.10

a = 0.09

a = 0.08

a = 0.07

a = 0.06

a = 0.05

a = 0.05

a = 0.06

a = 0.06

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

a = 0.057

a = 0.056

a = 0.055

a = 0.054

a = 0.053

What happens for even smaller values of alpha?

a = 0.05

a = 0.048

a = 0.046

a = 0.044

a = 0.042

a = 0.040

a = 0.040

a = 0.040

a = 0.039

a = 0.039

a = 0.030

a = 0.020

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

a = 0.039

a = 0.1

a = 0.8

a = 0.82

a = 0.823

a = 0.823

a = 0.824

Stability of the (nonlinear) modes

Stability of the (nonlinear) modes In this region both modes are stable

a = 0.07 Initial conditions: z1(0) = 0.1, z2(0) = 0.00209816, other i.c. are zero

a = 0.07

a = 0.07

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

Locking + entrainment in space of drive amplitude and coupling strength

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

Summary We have presented a simplified model of a MEMS oscillator, and we have investigated the dynamics of a system of two such coupled oscillators . Acknowledgement This work was supported by NSF grant ACI-1548562 NSF grant CMMI 1634664

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