Abstract
This paper considers nonlinear interactions between vibration modes with a focus on recent studies relevant to micro- and nanoscale mechanical resonators. Due to their inherently small damping and high susceptibility to nonlinearity, these devices have brought to light new phenomena and offer the potential for novel applications. Nonlinear interactions between vibration modes are well known to have the potential for generating a “zoo” of complicated bifurcation patterns and a wide variety of dynamic behaviors, including chaos. Here, we focus on more regular, robust, and predictable aspects of their dynamics, since these are most relevant to applications. The investigation is based on relatively simple two-mode models that are able to capture and predict a wide range of transient and sustained dynamical behaviors. The paper emphasizes modeling and analysis that has been done in support of recent experimental investigations and describes in full detail the analysis and attendant insights obtained from the models that are briefly described in the experimental papers. Standard analytical tools are employed, but the questions posed and the conclusions drawn are novel, as motivated by observations from experiments. The paper considers transient dynamics, response to harmonic forcing, and self-excited systems and describes phenomena such as extended coherence time during transient decay, zero dispersion response, and nonlinear frequency veering. The paper closes with some suggested directions for future studies in this area.
Highlights
Vibratory mechanical structures of microscale dimensions are ubiquitous in modern sensing and timekeeping technologies and are used in virtually every smart phone, tablet, automobile, and more [1,2]
If the degree of freedom (DOF) coordinates employed for the model are linearly coupled, the first step is to decouple the system at linear order by converting to the eigenmode coordinates
We note that the additional conservative restoring force changes sign as the frequency mismatch Δω2 is swept through zero during decay, which means that on one side of the internal resonance (IR) condition, where Δω2 > 0, it acts as a stiffening nonlinearity, while on the other side, where Δω2 < 0, it acts as a softening nonlinearity
Summary
Vibratory mechanical structures of microscale dimensions are ubiquitous in modern sensing and timekeeping technologies and are used in virtually every smart phone, tablet, automobile, and more [1,2]. Some of the nonlinear phenomena observed in MNRs, especially related to modal interactions, had not been previously seen in vibrations of macroscale structures, primarily because MNRs have significantly smaller damping Such extremely light damping, with damping ratios as small as 10−8 at room temperature [7], is desirable for many sensing and time-keeping applications, where devices operate near resonance and require high frequency selectivity, that is, a very sharp resonance peak. In this approach, which requires some experience, if one is trying to compare model results to experimental measurements, coefficients are obtained by fitting with experimental data This is straightforward for linear resonators operating in a single mode since one needs to know only the natural frequency, damping, and a parameter that captures the susceptibility of the device to applied forces, that is, an effective modal mass or stiffness.
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