Research on Internal Resonance Mechanisms in Micromechanical Resonators and Their Application in Frequency Stabilization

Background Introduction

Micromechanical resonators play a crucial role in modern timekeeping and sensing devices due to their high frequency, high quality factor, and high sensitivity. However, the extremely low damping characteristics of these resonators can lead to various nonlinear phenomena, thereby affecting their frequency stability. Among these, the Duffing hardening effect is a major limiting factor, causing frequency drift through amplitude variations, known as the amplitude-frequency effect. In recent years, internal resonance (InRes) has been proposed as an effective method to mitigate this issue and enhance frequency stability. Internal resonance facilitates energy exchange between vibrational modes with commensurate frequency ratios through nonlinear coupling terms, thereby achieving frequency stabilization.

To gain a deeper understanding of the role of internal resonance in frequency stabilization, researchers conducted a theoretical analysis of 1:2 and 1:3 internal resonance mechanisms, exploring the effects of different parameters on frequency stabilization. This study not only provides valuable guidance for designing micromechanical resonators with high-frequency stability but also reveals the potential of internal resonance in practical applications.

Source of the Paper

This paper was co-authored by Ata Donmez, Hansaja Herath, and Hanna Cho, all affiliated with the Department of Mechanical and Aerospace Engineering at The Ohio State University. The paper was accepted on February 8, 2025, and published in the journal Nonlinear Dynamics. The research was partially funded by the National Science Foundation (NSF-CMMI-2227527) and the Gear and Power Transmission Consortium at Ohio State University.

Research Process and Results

Theoretical Model and Methodology

The study employed a generalized two-mode reduced-order model, which includes Duffing nonlinearity and nonlinear modal coupling terms. By analyzing frequency response curves (FRCs) and p/2-backbone curves, the researchers demonstrated the effects of different parameters on frequency stabilization.

Model Construction

1. Equation Establishment: The study established equations of motion based on the externally driven mode u1 and the internally resonant higher mode u2. The u1 mode is excited by an external driving force f(t), while the u2 mode interacts with the u1 mode through internal coupling terms.
2. Steady-State Solution Assumption: The steady-state solutions of the system were assumed to be harmonic, and the nonlinear algebraic equations were solved using the harmonic balance method.
3. Stability Analysis: The stability of periodic solutions was analyzed using Floquet multipliers, and bifurcation behaviors were identified.

Parameter Study and Frequency Stabilization Mechanisms

Through systematic analysis of different parameters (such as coupling strength and frequency mismatch), the study revealed two distinct frequency stabilization mechanisms: weak coupling and strong coupling.

Weak Coupling Mechanism

In the weak coupling regime, frequency stabilization is achieved through amplitude and frequency saturation for both 1:2 and 1:3 internal resonance. As the driving amplitude increases, the frequency response curve separates near the internal resonance frequency, forming an isolated branch (isola), thereby achieving frequency and amplitude saturation over a certain driving range.

Strong Coupling Mechanism

In the strong coupling regime, 1:2 internal resonance reduces the amplitude-frequency effect by forming an asymptote, while 1:3 internal resonance achieves frequency stabilization through a zero-dispersion point. Strong coupling significantly alters the shape of the p/2-backbone curve, enabling frequency stabilization over a broader driving range.

Experimental Validation and Numerical Simulation

The study validated the accuracy of the theoretical model through numerical simulations and compared the results with experimental data. The numerical simulation results showed that the theoretical model could effectively predict the dynamic behavior of the system, particularly the frequency stabilization phenomenon near the internal resonance frequency.

Research Conclusions

Through theoretical analysis and numerical simulations, the study thoroughly investigated the role of 1:2 and 1:3 internal resonance in frequency stabilization of micromechanical resonators. The research found that internal resonance, through modal coupling and nonlinear interactions, can effectively suppress the amplitude-frequency effect, thereby achieving frequency stabilization. The specific conclusions are as follows: 1. Weak Coupling Mechanism: In the weak coupling regime, frequency stabilization is achieved through amplitude and frequency saturation, applicable to both 1:2 and 1:3 internal resonance. 2. Strong Coupling Mechanism: In the strong coupling regime, 1:2 internal resonance stabilizes frequency through an asymptote, while 1:3 internal resonance achieves frequency stabilization via a zero-dispersion point. 3. Parameter Influence: Coupling strength and frequency mismatch are key parameters determining the effectiveness of frequency stabilization, and appropriate parameter selection can significantly enhance resonator frequency stability.

Research Highlights

1. Theoretical Innovation: The study proposed a generalized two-mode reduced-order model that effectively describes the influence of internal resonance and nonlinear interactions on frequency stabilization.
2. Application Value: The research results provide theoretical guidance for designing micromechanical resonators with high-frequency stability, offering significant engineering application value.
3. Experimental Validation: Through numerical simulations and experimental validation, the study confirmed the accuracy of the theoretical model, laying the foundation for subsequent experimental research.

Other Valuable Information

The study also explored the potential applications of internal resonance in micromechanical sensors. By eliminating the amplitude-frequency effect in the nonlinear regime, internal resonance can significantly extend the dynamic range of sensors, improving the signal-to-noise ratio. Additionally, the study proposed the possibility of generating frequency combs using internal resonance, opening new avenues for advanced sensing and signal processing.

Summary

Through theoretical analysis and numerical simulations, this study thoroughly investigated the role of 1:2 and 1:3 internal resonance in frequency stabilization of micromechanical resonators. The research results not only provide theoretical guidance for designing resonators with high-frequency stability but also open new research directions for the application of internal resonance in sensing and signal processing. Future experimental studies will further validate and extend the theoretical findings of this research, promoting the widespread application of micromechanical resonators in engineering fields.