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<p>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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_10987_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">\(\pi /2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>π</mi> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>–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.</p>

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Theoretical insights into 1:2 and 1:3 internal resonance for frequency stabilization in nonlinear micromechanical resonators

  • 期刊论文

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  • 发表日期：

    2025年3月11日

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  • Ata Donmez,
  • Hansaja Herath,
  • Hanna Cho

摘要

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 \(\pi /2\) π / 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.