%ARand, R%AShayak, B%ABhaskar, A%AZehnder, A.%BJournal Name: International journal of engineering research and applications; Journal Volume: 11; Journal Issue: 10
%D2020%I
%JJournal Name: International journal of engineering research and applications; Journal Volume: 11; Journal Issue: 10
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%MOSTI ID: 10309647
%PMedium: X
%TDynamics of a system of two coupled third-order MEMS oscillators
%XIn this work we present a systematic review of novel and interesting behaviour we have observed in a simplified
model of a MEMS oscillator. The model is third order and nonlinear, and we expressit as a single ODE for a
displacement variable. We find that a single oscillator exhibits limitcycles whose amplitude is well
approximated by perturbation methods. Two coupled identicaloscillators have in-phase and out-of-phase modes
as well as more complicated motions.Bothof the simple modes are stable in some regions of the parameter space
while the bifurcationstructure is quite complex in other regions. This structure is symmetric; the symmetry is
brokenby the introduction of detuning between the two oscillators. Numerical integration of the fullsystem is
used to check all bifurcation computations.
Each individual oscillator is based on a MEMS structure which moves within a laser-driven interference pattern.
As the structure vibrates, it changes the interference gap, causing the quantity of absorbed light to change,
producing a feedback loop between the motion and the absorbed light and resulting in a limit cycle oscillation.
A simplified model of this MEMS oscillator, omitting parametric feedback and structural damping, is
investigated using Lindstedt's perturbation method. Conditions are derived on the parameters of the model for a
limit cycle to exist.
The original model of the MEMS oscillator consists of two equations: a second order ODE which describes the
physical motion of a microbeam, and a first order ODE which describes the heat conduction due to the laser.
Starting with these equations, we derive a single governing ODE which is of third order and which leads to the
definition of a linear operator called the MEMS operator. The addition of nonlinear terms in the model is shown
to produce limit cycle behavior.
The differential equations of motion of the system of two coupled oscillators are numerically integrated for
varying values of the coupling parameter. It is shown that the in-phase mode loses stability as the coupling
parameter is reduced below a certain value, and is replaced by two new periodic motions which are born in a
pitchfork bifurcation. Then as this parameter is further reduced, the form of the bifurcating periodic motions
grows more complex, with yet additional bifurcations occurring. This sequence of bifurcations leads to a
situation in which the only periodic motion is a stable out-of-phase mode. The complexity of the resulting
sequence of bifurcations is illustrated through a series of diagrams based on numerical integration.
%0Journal Article