SNIC bifurcation and its Application to MEMS

TL;DR

This paper presents a method to generate frequency combs in MEMS devices by inducing SNIC bifurcation through non-linear beam vibrations and external harmonic perturbation, supported by theoretical and numerical analysis.

Contribution

It introduces a novel approach to produce frequency combs in MEMS using SNIC bifurcation analysis and simplified modeling based on the Adler equation.

Findings

Theoretical model predicts frequency comb generation at SNIC bifurcation.

Numerical simulations confirm the validity of the simplified Adler-based model.

The method demonstrates practical potential for MEMS frequency comb applications.

Abstract

This project focuses on a method to extract a frequency comb in mechanical means, for general interest and numerous practical applications in MEMS. The method of execution is the implementation of a beam that is exhibiting non-linear dynamics that is perturbed and analyzed for its transverse vibrations. The perturbation is an external harmonic driver with a chosen small amplitude and frequency (which is slightly detuned from the beam eigenfrequency), that when engaged with the unperturbed beam oscillations, causes it reach a state of "injection pulling" - an effect that occurs when one harmonic oscillator is coupled with a second one and causes it to oscillate in a frequency near its own. This causes the beam to reach SNIC bifurcation, rendering a frequency comb as desired. Theoretical analysis showed that the problem can be modelled using a non-linear equation of the beam, that translates to a form of the non-linear Duffing equation. While a solution to the dynamics function of the beam is hard to obtain in practice due to mathematical difficulties, a slow evolution model is suggested that is composed of functions of a amplitude and phase. Using several additional mathematical assumptions, the amplitude is seen to be related to the phase, while the phase equation solution is seen to be of the form of Adler's equation. These assumptions ultimately reduce the entire behaviour of the beam to a relatively simple solution to the Adler equation, which has a known analytical solution. Computerized numerical simulations are run on it to check the results and compare them to the theory and desired outcome. The results agreed with the theory and produce the expected frequency comb, showing the assumptions to be valid in extracting the comb.