Dynamics of MEMS Resonators

Optically Transduced MEMS:

Our group fabricates silicon doubly and singly supported beams suspended over a silicon substrate using photo-lithography, and illuminates them with a continuous wave laser. For low laser power, the beams bend statically, but beyond a threshold laser power the beams self-oscillate. Such self-sustained oscillations are called limit cycle oscillations (LCO), and have been observed in systems as varied as oil whip in hydrodynamic bearings, aero-elastic flutter in airplane wings, and thermally induced oscillations in satellites.

Array of singly and doubly clamped beams fabricated form SOI wafers
Array of singly and doubly clamped beams fabricated form SOI wafers

optical_transduction

Understanding the Causes of LCO:

It is known that thermally induced limit cycle oscillations are driven by coupling between heating and displacement, and yet a simple beam theory model of a doubly supported beam subject to centerline heating shows no displacement until buckling. In order to understand the thermal-mechanicaLCO_thermal_drivel coupling in our beams, we built and Finite Element Method (FEM) model of a doubly supported beam subject to centerline heating. The model showed that displacement due to heating is caused by asymmetry of the support which leads to a phenomenon known as “imperfection buckling.”

Analysis of Model Equations:

We model the temperature and displacement of these thermally actuated beams using coupled non-linear differential equations and use a numerical continuation software package, AUTO2000, to analyze the equilibrium (i.e. static deflection) solutions and stable periodic (i.e. limit cycle) solutions. Analysis of the equations suggests that periodicity in the interference field should lead to multiple limit cycle oscillations with periodically spaced amplitudes. Such models also allow us to predict the threshold power for self-oscillation.

multiple_LCO

Limit Cycle Oscillation and Entrainment:

For high laser power, we can entrain the limit cycle by inertially driving the beam. In addition to 1:1 entrainment we observe sub-harmonic entrainment of order 2:1 up to 7:1 as well as super-harmonic entrainment of orders 1:2 and 1:3. We find that when entrained, the amplitude of response follows the backbone curve, and that the asymmetry of this curve leads to asymmetry of the regions of entrainment. To our knowledge sub- and super-harmonic entrainments of this order have not been previously observed in MEMS. Sub- and super-harmonic entrainment could be used for signal division or multiplication.

entrainmenttunability

Acknowledgements

This work was supported under NSF grant 0600174 and performed in part at the Cornell NanoScale Facility, a member of the National Nanotechnology Infrastructure Network, which is supported by the National Science Foundation (Grant ECS-0335765). This work also made use of the Integrated Advanced Microscopy and Materials facilities of the Cornell Center for Materials Research (CCMR) with support from the NSF Materials Research Science and Engineering Centers (MRSEC) program (DMR 1120296).