Theoretical Insights into 1:2 and 1:3 Internal Resonance for Frequency Stabilization in Nonlinear Micromechanical Resonators

TL;DR

This paper provides a theoretical analysis of internal resonance phenomena in nonlinear micromechanical resonators, revealing how 1:2 and 1:3 InRes can be exploited to improve frequency stability by understanding their nonlinear dynamics.

Contribution

It introduces a generalized two-mode model incorporating Duffing nonlinearity to analyze internal resonance effects on frequency stabilization in micromechanical resonators.

Findings

Weak coupling leads to amplitude and frequency saturation for stabilization.

Strong coupling creates asymptotes or zero-dispersion points reducing amplitude-frequency effects.

Insights guide design of resonators with enhanced frequency stability.

Abstract

Micromechanical resonators are essential components in time-keeping and sensing devices due to their high frequency, high quality factor, and sensitivity. However, their extremely low damping can lead to various nonlinear phenomena that can compromise frequency stability. A major limiting factor is the Duffing hardening effect, which causes frequency drift through amplitude variations, known as the amplitude-frequency effect. Recently, internal resonance (InRes) has emerged as an effective approach to mitigate this issue and enhance frequency stabilization. In this study, we investigate the frequency stabilization mechanisms of 1:2 and 1:3 InRes using a generalized two-mode reduced-order model that includes Duffing nonlinearity and nonlinear modal coupling. By analyzing the frequency response curves and pi/2-backbone curves, we demonstrate how different parameters affect the effectiveness of frequency stabilization. Our results identify two distinct regimes depending on the coupling strength relative to the stiffening effect as a key factor in determining the stabilization mechanism. For the regime of weak coupling, both 1:2 and 1:3 InRes achieve frequency stabilization through amplitude and frequency saturation over a range of forcing amplitudes. In contrast, strong coupling reduces the amplitude-frequency effect by forming an asymptote line for 1:2 InRes or a zero-dispersion point for 1:3 InRes. These insights offer valuable guidelines for designing micromechanical resonators with high-frequency stability, highlighting InRes as a robust tool for enhancing performance in practical applications.